Chapter 16 : Probability - Rd Sharma Solutions for Class 10 Maths CBSE

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Chapter 16 - Probability Exercise Ex. 16.1

Question 1

Solution 1

Question 2

A die is thrown. Find the probability of getting:

(i) a prime number

(ii) 2 or 4

(iii) a multiple of 2 or 3

(iv) an even prime number

(v) a number greater than 5

(iv) a number lying between 2 and 6

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is neither a king nor a queen.

Solution 7

 

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

If the probability of winning a game is 0.3, what is the probability of loosing it?

Solution 16

Question 17

A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

(i) red (ii) black or white (iii) not black

Solution 17

Question 18

A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

(i) white (ii) red (iii) not black (iv) red or white

Solution 18

Question 19

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red suit

(ii) a face card

(iii) a red face card

(iv) a queen of black suit

(v) a jack of hearts

(vi) a spade

Solution 19

Question 20

Five cards-ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random.

(i) What is the probability that the card is a queen?


Solution 20

open parentheses straight i close parentheses space Total space number space of space cards space equals space 5 Total space number space of space space queen space equals space 1 straight P open parentheses getting space straight a space queen close parentheses equals fraction numerator Number space of space favourable space outcomes over denominator Total space number space of space outcomes end fraction equals 1 fifth

Question 21

Five cards – ten, jack, queen, king, and an ace of diamonds are shuffled face downwards.  One card is picked at random.

 

If a king is drawn first and put aside, what is the probability that the second card picked up is the

(i)ace? (ii)king?

Solution 21

table attributes columnalign left end attributes row cell S i n c e text    end text t h e text    end text f i r s t text    end text c a r d text    end text d r a w n text    end text i s text    end text a text    end text k i n g comma text    end text t h e r e f o r e text    end text t h e r e text    end text a r e end cell row cell 4 text    end text r e m a i n i n g text    end text c a r d s. end cell row cell left parenthesis i right parenthesis n left parenthesis t o t a l text    end text o u t c o m e s right parenthesis equals 4 end cell row cell n left parenthesis f a v o u r a b l e text    end text o u t c o m e right parenthesis equals 1 end cell row cell P left parenthesis s e c o n d text    end text c a r d text    end text i s text    end text a n text    end text a c e right parenthesis equals 1 fourth end cell row blank row cell S i n c e text    end text t h e text    end text f i r s t text    end text c a r d text    end text d r a w n text    end text i s text    end text a text    end text k i n g comma text    end text t h e r e f o r e text    end text t h e r e text    end text a r e end cell row cell 4 text    end text r e m a i n i n g text    end text c a r d s text    end text a n d text    end text t h e r e text    end text a r e text    end text n o text    end text r e m a i n i n g text     end text k i n g s text   end text. end cell row cell left parenthesis i i right parenthesis n left parenthesis t o t a l text    end text o u t c o m e s right parenthesis equals 5 end cell row cell n left parenthesis f a v o u r a b l e text    end text o u t c o m e right parenthesis equals 0 end cell row cell P left parenthesis s e c o n d text    end text c a r d text    end text i s text    end text a text    end text k i n g right parenthesis equals 0 over 5 equals 0 end cell end table

Question 22

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red? (ii) black?

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

A coin has only two options-head and tail and both are equally likely events i.e. the probability of occurrence of both is same. Hence, a coin is a fair option to decide which team will choose ends in the game.

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Harpreet tosses two different coins simultaneously (say, one is of Re 1 and other of Rs 2). What is the probability that gets at least one head?

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Fill in the blanks:

(i) Probability of a sure event is ________.

(ii) Probability of an impossible event is _______.

(iii) The probability of an event (other than sure and impossible event) lies between ______.

(iv) Every elementary event associated to a random experiment has _______ probability.

(v) Probability of an event A + Probability of an event 'not A' = ___________.

(vi) Sum of the probabilities of each outcome in an experiment is ____________.

Solution 43

(i) 1

(ii) 0

(iii) 0 and 1

(iv) equal

(v) 1

(vi) 1

Question 44

Solution 44

Question 45

A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the probability that it will be (i) a blue card (ii) not a yellow card (iii) neither yellow nor a blue card.

Solution 45

Question 46

A box contains cards numbered 3,5,7,9,…,35,37. A card is drawn at random from the box. Find the probability that the number on the drawn card is a prime number.

Solution 46

table attributes columnalign left end attributes row cell N u m b e r e d text   cards  =  end text left curly bracket text 3 ,  5 ,  7 ,  9 ,  11 ,  13 ,  15 ,  17 ,  19 ,  21 ,  23 ,  25 ,  27 ,  29 ,  31 ,  33 ,  35 ,  37 end text right curly bracket end cell row cell n left parenthesis t o t a l text    end text n u m b e r text    end text o f text    end text o u t c o m e s right parenthesis equals 18 end cell row cell N u m b e r text   of   prime   cards = end text left curly bracket text 3 ,  5 ,  7 ,  11 ,  13 ,  17 ,  19 ,  23 ,  29 ,  31 ,  37 end text right curly bracket end cell row cell n left parenthesis f a v o u r a b l e text    end text o u t c o m e s right parenthesis equals 11 end cell row blank row cell P left parenthesis n u m b e r text    end text d r a w n text    end text i s text    end text a text    end text p r i m e text    end text n u m b e r right parenthesis equals 11 over 18 end cell row blank row cell asterisk times N o t e colon text   Answer   given   in   the   book   is   incorrect. end text end cell end table

Question 47

A group consists of 12 persons, of which 3 are extremely patient, 6 are extremely honest and the rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest. Which of the above would you prefer more?

Solution 47

 

 

table attributes columnalign left end attributes row cell n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text p e r s o n s right parenthesis equals 12 end cell row cell n left parenthesis e x t r e m e l y text    end text p a t i e n t right parenthesis equals 3 end cell row cell n left parenthesis e x t r e m e l y text    end text h o n e s t right parenthesis equals 6 end cell row cell n left parenthesis e x t r e m e l y text    end text k i n d right parenthesis equals 3 end cell row cell left parenthesis i right parenthesis P left parenthesis e x t r e m e l y text    end text p a t i e n t right parenthesis equals fraction numerator n left parenthesis e x t r e m e l y text    end text p a t i e n t right parenthesis over denominator n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text p e r s o n s right parenthesis end fraction end cell row cell equals 3 over 12 equals 1 fourth end cell row cell left parenthesis i i right parenthesis P left parenthesis e x t r e m e l y text    end text k i n d text    end text o r text    end text h o n e s t right parenthesis equals fraction numerator n left parenthesis e x t r e m e l y text    end text k in d text    end text o r text    end text h o n e s t right parenthesis over denominator n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text p e r s o n s right parenthesis end fraction end cell row cell equals 6 over 12 plus 3 over 12 equals fraction numerator 6 plus 3 over denominator 12 end fraction equals 9 over 12 equals 3 over 4 end cell row cell I text   would   prefer   the   person   selected   to   be   kind   and   honest. end text end cell end table

 

 

 

Question 48

Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is,

(i) Not divisible by 3

(ii) A prime number greater than 7

(iii) Not a perfect square number.

Solution 48

Question 49

A piggy bank contains hundred 50 paise coins, fifty Rs. 1 coins, twenty Rs. 2 coins and ten Rs. 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability that the coin which falls out

(i) is a 50 paise win

(ii) is of value more than Rs. 1

(iii) is of value less than Rs. 5

(iv) is a Rs. 1 or Rs. 2 coin

Solution 49

 

table attributes columnalign left end attributes row cell n left parenthesis 50 text    end text p a i s a text    end text c o i n s right parenthesis equals 100 end cell row cell n left parenthesis R e.1 text    end text c o i n s right parenthesis equals 50 end cell row cell n left parenthesis R s.2 text    end text c o i n s right parenthesis equals 20 end cell row cell n left parenthesis R s.5 text    end text c o i n s right parenthesis equals 10 end cell row cell n left parenthesis T o t a l text    end text n o. text    end text o f text    end text c o i n s right parenthesis equals 180 end cell row blank row cell left parenthesis i right parenthesis P left parenthesis w i l l text    end text b e text    end text a text    end text 50 text    end text p a i s a text    end text w i n right parenthesis equals 100 over 180 equals 5 over 9 end cell row cell left parenthesis i i right parenthesis P left parenthesis w i l l text    end text b e text   end text o f text    end text v a l u e text    end text m o r e text    end text t h a n text    end text R e.1 right parenthesis equals 30 over 180 equals 1 over 6 end cell row cell left parenthesis i i i right parenthesis P left parenthesis w i l l text    end text b e text    end text o f text    end text v a l u e text    end text l e s s text    end text t h a n text    end text R s.5 right parenthesis equals 170 over 180 equals 17 over 18 end cell row cell left parenthesis i v right parenthesis P left parenthesis w i l l text    end text b e text    end text a text    end text R e.1 text    end text o r text    end text R s.2 text    end text c o i n s right parenthesis equals 70 over 180 equals 7 over 18 end cell end table

 

 

 

Question 50

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the card thoroughly. Find the probability that the number on the drawn card is

(i) an odd number

(ii) a multiple of 5

(iii) a perfect square

(iv) an even prime number

Solution 50

Question 51

A box contains 20 cards numbered from 1 to 20. A card is drawn at random from the box. Find the probability that the number on the drawn card is

  1. divisible 2 or 3
  2. a prime number
Solution 51

Question 52

 

left parenthesis x i v right parenthesis space A n space e v e n space n u m b e r space o n space e a c h space d i c e left parenthesis x v right parenthesis space 5 space a s space t h e space s u m

 

Solution 52

 

 
 
left parenthesis x i v right parenthesis table attributes columnalign left end attributes row cell text Consider   the   set   of   ordered   pairs end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row cell C l e a r l y comma text   there   are   36   elementary   events. end text end cell row cell therefore n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s right parenthesis equals 36 end cell row cell G e t t i n g text   an   even   number   on   each   dice   can   be   selected end text end cell row cell text as   listed   below : end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row blank row cell T h e r e f o r e comma text   end text n left parenthesis F a v o u r a b l e text    end text e v e n t s right parenthesis equals 9 end cell row blank row cell P left parenthesis g e t t i n g text    end text a n text    end text e v e n text    end text n u m b e r text    end text o n text    end text e a c h text    end text d i c e right parenthesis end cell row cell equals fraction numerator g e t t i n g text    end text e v e n text    end text n u m b e r text    end text o n text    end text e a c h text    end text d i c e over denominator T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s end fraction end cell row cell equals 9 over 36 end cell row cell equals 1 fourth end cell end table
 
 
left parenthesis x v right parenthesis table attributes columnalign left end attributes row cell text Consider   the   set   of   ordered   pairs end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row cell C l e a r l y comma text   there   are   36   elementary   events. end text end cell row cell therefore n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s right parenthesis equals 36 end cell row cell N u m b e r text   of   pairs   in   end text g e t t i n g text    end text 5 text   end text a s text   end text t h e text   end text s u m text   can   be   selected end text end cell row cell text as   listed   below : end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row blank row cell T h e r e f o r e comma text   end text n left parenthesis F a v o u r a b l e text    end text e v e n t s right parenthesis equals 4 end cell row blank row cell P left parenthesis g e t t i n g text    end text 5 text   as   the   sum end text right parenthesis end cell row cell equals fraction numerator n u m b e r text   of   pairs   in   end text g e t t i n g text    end text 5 text   end text a s text   end text t h e text   end text s u m over denominator T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s end fraction end cell row cell equals 4 over 36 end cell row cell equals 1 over 9 end cell end table
 
 
Question 53

In a simultaneous throw of a pair of dice, find the probability that:

2 will come up at least once

Solution 53

 

Question 54

In a simultaneous throw of a pair of dice, find the probability that:

2 will not come either time

Solution 54

 

Question 55

What is the probability that an ordinary year has 53 Sundays?

Solution 55

Question 56

Solution 56

Question 57

(viii) that the product of numbers appearing on the top of the dice is less than 9.

(ix) that the difference of the numbers appearing on the top of the two dice is 2.

(x) that the numbers obtained have a product less than 16.

left parenthesis v i i i right parenthesis A space b l a c k space d i c e space a n d space a space w h i t e space d i c e space a r e space t h r o w n space a t space t h e space s a m e space t i m e. space W r i t e space a l l space t h e space p o s s i b l e space o u t c o m e s. space W h a t space i s space t h e space p r o b a b i l i t y space t h a t space t h e space p r o d u c t space o f space t h e space n u m b e r s space a p p e a r i n g space o n space t h e space t o p space o f space t h e space d i c e space i s space l e s s space t h a n space 9 ?

left parenthesis i x right parenthesis A space b l a c k space d i c e space a n d space a space w h i t e space d i c e space a r e space t h r o w n space a t space t h e space s a m e space t i m e. space W r i t e space a l l space t h e space p o s s i b l e space o u t c o m e s. space W h a t space i s space t h e space p r o b a b i l i t y space t h a t space t h e space d i f f e r e n c e space o f space t h e space n u m b e r s space a p p e a r i n g space o n space t h e space t o p space o f space t h e space t w o space d i e space i s space 2.

Solution 57



left parenthesis v i i i right parenthesis table attributes columnalign left end attributes row cell text Consider   the   set   of   ordered   pairs end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row cell C l e a r l y comma text   there   are   36   elementary   events. end text end cell row cell therefore n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s right parenthesis equals 36 end cell row cell text Number   of   pairs   such   that   the   product   of   the   end text end cell row cell text number   on   each   dice   is   less   than   9   are end text end cell row cell text as   listed   below : end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row blank row cell T h e r e f o r e comma text   end text n left parenthesis F a v o u r a b l e text    end text e v e n t s right parenthesis equals 16 end cell row cell n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text o u t c o m e s right parenthesis equals 36 end cell row blank row cell P left parenthesis g e t t i n g text    end text 9 text   as   the   product end text right parenthesis end cell row cell equals fraction numerator n u m b e r text   of   pairs   in   end text g e t t i n g text    end text 9 text   end text a s text   end text t h e text   end text p r o d u c t over denominator T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s end fraction end cell row cell equals 16 over 36 end cell row cell equals 4 over 9 end cell row blank row blank row cell P left parenthesis P r o d u c t text    end text o f text    end text t h e text    end text n u m b e r s text    end text a p p e a r i n g text    end text o n text    end text t h e text    end text t o p text    end text o f text    end text t h e text   end text end cell row cell d i c e text    end text i s text    end text l e s s text    end text t h a n text    end text 9 right parenthesis equals 4 over 9 end cell end table

 

left parenthesis i x right parenthesis table attributes columnalign left end attributes row cell text Consider   the   set   of   ordered   pairs end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row cell C l e a r l y comma text   there   are   36   elementary   events. end text end cell row cell therefore n left parenthesis T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s right parenthesis equals 36 end cell row cell text Number   of   pairs   such   that   difference   of   the   numbers   end text end cell row cell text appearing   on   top   of   the   two   die   is   2   can   be   selected end text end cell row cell text as   listed   below : end text end cell row cell open curly brackets table row cell left parenthesis 1 comma 1 right parenthesis end cell cell left parenthesis 1 comma 2 right parenthesis end cell cell left parenthesis 1 comma 3 right parenthesis end cell cell left parenthesis 1 comma 4 right parenthesis end cell cell left parenthesis 1 comma 5 right parenthesis end cell cell left parenthesis 1 comma 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma 2 right parenthesis end cell cell left parenthesis 2 comma 3 right parenthesis end cell cell left parenthesis 2 comma 4 right parenthesis end cell cell left parenthesis 2 comma 5 right parenthesis end cell cell left parenthesis 2 comma 6 right parenthesis end cell row cell left parenthesis 3 comma 1 right parenthesis end cell cell left parenthesis 3 comma 2 right parenthesis end cell cell left parenthesis 3 comma 3 right parenthesis end cell cell left parenthesis 3 comma 4 right parenthesis end cell cell left parenthesis 3 comma 5 right parenthesis end cell cell left parenthesis 3 comma 6 right parenthesis end cell row cell left parenthesis 4 comma 1 right parenthesis end cell cell left parenthesis 4 comma 2 right parenthesis end cell cell left parenthesis 4 comma 3 right parenthesis end cell cell left parenthesis 4 comma 4 right parenthesis end cell cell left parenthesis 4 comma 5 right parenthesis end cell cell left parenthesis 4 comma 6 right parenthesis end cell row cell left parenthesis 5 comma 1 right parenthesis end cell cell left parenthesis 5 comma 2 right parenthesis end cell cell left parenthesis 5 comma 3 right parenthesis end cell cell left parenthesis 5 comma 4 right parenthesis end cell cell left parenthesis 5 comma 5 right parenthesis end cell cell left parenthesis 5 comma 6 right parenthesis end cell row cell left parenthesis 6 comma 1 right parenthesis end cell cell left parenthesis 6 comma 2 right parenthesis end cell cell left parenthesis 6 comma 3 right parenthesis end cell cell left parenthesis 6 comma 4 right parenthesis end cell cell left parenthesis 6 comma 5 right parenthesis end cell cell left parenthesis 6 comma 6 right parenthesis end cell end table close curly brackets end cell row blank row cell T h e r e f o r e comma text   end text n left parenthesis F a v o u r a b l e text    end text e v e n t s right parenthesis equals 8 end cell row blank row cell P left parenthesis text difference   of   the   numbers   is   2 end text right parenthesis end cell row cell equals fraction numerator table attributes columnalign left end attributes row cell text Number   of   pairs   such   that   difference   of   the   numbers   end text end cell row cell text Appearing   on   the   top   of   the   two   dice   is   2   end text end cell end table over denominator T o t a l text    end text n u m b e r text    end text o f text    end text t h r o w s end fraction end cell row cell equals 8 over 36 end cell row cell equals 2 over 9 end cell end table

 

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

(iv) card  (v) diamond

Solution 65

Question 66

Solution 66

Question 67

A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

Solution 67

Question 68

A dice is rolled twice. Find the probability that

(i) 5 will not come up either time.

(ii) 5 will come up exactly once.

Solution 68

Question 69

All the black face cards are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a

(i) Face card

(ii) Red card

(iii) Black card

(iv) King

Solution 69

 

 

table attributes columnalign left end attributes row cell T h e text   number   of   black   face   cards  =  6 end text end cell row cell S i n c e comma a l l text    end text t h e text    end text b l a c k text   end text f a c e text   end text c a r d s text    end text h a v e text    end text b e e n text    end text r e m o v e d comma t h e r e f o r e end cell row cell t h e text    end text r e m a i n i n g text    end text n u m b e r text    end text o f text    end text c a r d s equals 52 minus 6 equals 46 end cell row cell n left parenthesis t o t a l text    end text n u m b e r text    end text o f text    end text c a r d s text    end text r e m a i n i n g right parenthesis equals 46 end cell row blank row cell left parenthesis i right parenthesis T o t a l text   number   of   face   cards  =  6  +  6  =  12 end text end cell row cell W h e n text   removing   all   black   face   cards ,  there   are   6   red   face   cards. end text end cell row cell rightwards double arrow n left parenthesis g e t t i n g text    end text a text    end text f a c e text    end text c a r d right parenthesis equals 6 end cell row cell rightwards double arrow P left parenthesis g e t t i n g text    end text a text    end text f a c e text    end text c a r d right parenthesis equals 6 over 46 equals 3 over 23 end cell row cell left parenthesis i i right parenthesis T o t a l text   number   of   red   card = 26   end text end cell row cell rightwards double arrow n left parenthesis g e t t i n g text    end text a text    end text r e d text    end text c a r d right parenthesis equals 26 end cell row cell rightwards double arrow P left parenthesis g e t t i n g text    end text a text    end text r e d text    end text c a r d right parenthesis equals 26 over 46 equals 13 over 23 end cell row cell left parenthesis i i i right parenthesis N u m b e r text   of   black   face   cards  =  6 end text end cell row cell text Total   number   of   black   cards  =  26 end text end cell row cell text Remaining   number   of   black   cards = 26 end text minus 6 equals 20 end cell row cell rightwards double arrow n left parenthesis g e t t i n g text    end text a text    end text b l a c k text    end text c a r d right parenthesis equals 20 end cell row cell rightwards double arrow P left parenthesis g e t t i n g text    end text a text    end text b l a c k text    end text c a r d right parenthesis equals 20 over 46 equals 10 over 23 end cell row cell left parenthesis i v right parenthesis N u m b e r text   of   black   king   cards  =  2 end text end cell row cell text Total   number   of   king   cards  =  4 end text end cell row cell text Remaining   number   of   red   king   cards  =  2 end text end cell row cell rightwards double arrow n left parenthesis g e t t i n g text    end text a text    end text k i n g right parenthesis equals 2 end cell row cell rightwards double arrow P left parenthesis g e t t i n g text    end text a text    end text k i n g right parenthesis equals 2 over 46 equals 1 over 23 end cell end table

 

 

Question 70

Cards numbered from 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn cards is

(i) an odd number

(ii) a perfect square number

(iii) divisible by 5

(iv) a prime number less than 20

Solution 70

Question 71

All kings and queens are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is randomly drawn from it. Find the probability that this card is

(i) a red face card

(ii) a black card

Solution 71

table attributes columnalign left end attributes row cell F a c e text   end text c a r d s text   queen   and   king ,  that   is   4   end text q u e e n s text    end text end cell row cell a n d text    end text 4 text    end text k i n g s text    end text a r e text    end text r e m o v e d. end cell row cell T h u s comma text   in   total   4  +  4  =  8   cards   are   removed. end text end cell row cell T h e text    end text r e m a i n i n g text   end text n u m b e r text    end text o f text    end text c a r d s equals 52 minus 8 equals 44 end cell row cell n left parenthesis t o t a l text    end text r e m a n i n g text    end text c a r d s right parenthesis equals 44 end cell row cell left parenthesis i right parenthesis end cell row cell T o t a straight l text   number   of   face   cards  =  6  +  6  =  12 end text end cell row cell text Number   of   red   face   cards  =  2  +  2  +  2  =  6 end text end cell row cell W h e n text   we   remove   four   red   face   cards   queen   and   king , end text end cell row cell text two   jack   red   face   cards   are   remaining. end text end cell row cell rightwards double arrow n left parenthesis a text    end text r e d text    end text f a c e text    end text c a r d right parenthesis equals 2 end cell row cell rightwards double arrow P left parenthesis a text    end text r e d text    end text f a c e text    end text c a r d right parenthesis equals 2 over 44 equals 1 over 22 end cell row cell left parenthesis i i right parenthesis end cell row cell text When   we   remove   four   black   face   cards   queen   and   king ,  end text end cell row cell text the   number   of   balck   cards   is   26 end text minus text 4 = 22 end text end cell row cell rightwards double arrow n left parenthesis a text    end text b l a c k text    end text c a r d right parenthesis equals 22 end cell row cell rightwards double arrow P left parenthesis a text    end text b l a c k text    end text c a r d right parenthesis equals 22 over 44 equals 1 half end cell end table

Question 72

All jacks, queens and kings are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is randomly drawn from it. Find the probability that this card is

(i) a black face card

(ii) a red card

Solution 72

table attributes columnalign left end attributes row cell F a c e text   end text c a r d s text   are   jack ,  queen   and   king. end text end cell row cell 4 text   end text j a c k s comma text   4   end text q u e e n s text    end text a n d text    end text 4 text    end text k i n g s text    end text a r e text    end text r e m o v e d. end cell row cell T h u s comma text   in   total   4  +  4  +  4  =  12   cards   are   removed. end text end cell row cell T h e text    end text r e m a i n i n g text   end text n u m b e r text    end text o f text    end text c a r d s equals 52 minus 12 equals 40 end cell row cell n left parenthesis t o t a l text    end text r e m a n i n g text    end text c a r d s right parenthesis equals 40 end cell row blank row cell left parenthesis i right parenthesis end cell row cell W h e n text   we   remove   all   the   face   cards ,  both   black   and end text end cell row cell text red ,  the   number   of   black   face   cards   is   zero. end text end cell row cell rightwards double arrow n left parenthesis b l a c k text    end text f a c e text    end text c a r d right parenthesis equals 0 end cell row cell rightwards double arrow P left parenthesis b l a c k text    end text f a c e text    end text c a r d right parenthesis equals 0 over 40 equals 0 end cell row cell left parenthesis i i right parenthesis end cell row cell T h e r e text   are   26   red   cards   and   in   that    there   are   6   face   cards end text end cell row cell text When   we   remove   all   the   face   cards ,  the   number   of end text end cell row cell text red   cards   is   26 end text minus text 6 = 20 end text end cell row cell rightwards double arrow n left parenthesis a text    end text r e d text    end text c a r d right parenthesis equals 20 end cell row cell rightwards double arrow P left parenthesis a text    end text r e d text    end text c a r d right parenthesis equals 20 over 40 equals 1 half end cell end table

Question 73

Red queens and black jacks are removed from a pack of 52 playing  is drawn at random from the remaining cards, after reshuffling them. Find the probability that the card drawn is

(i) a king

(ii) of red colour

(iii) a face card

(iv) a queen

Solution 73

 

table attributes columnalign left end attributes row cell S i n c e text    end text r e d text    end text q u e e n s text    end text a n d text    end text b l a c k text    end text c a r d s text    end text a r e text    end text r e m o v e d text    end text f r o m text    end text a text    end text p a c k text    end text o f end cell row cell 52 text    end text c a r d s comma r e m a i n i n g text    end text c a r d s equals 44 minus 4 equals 48 end cell row cell n left parenthesis t o t a l text    end text n u m b e r text    end text o f text    end text c a r d s right parenthesis equals 48 end cell row cell n left parenthesis a text    end text k i n g right parenthesis equals 4 end cell row cell n left parenthesis o f text    end text r e d text    end text c o l o u r right parenthesis equals 24 end cell row cell n left parenthesis a text    end text f a c e text    end text c a r d right parenthesis equals 8 end cell row cell n left parenthesis a text    end text q u e e n right parenthesis equals 2 end cell row cell left parenthesis i right parenthesis P left parenthesis a text    end text k i n g right parenthesis equals text   end text 4 over 48 equals 1 over 12 end cell row cell left parenthesis i i right parenthesis P left parenthesis o f text    end text r e d text    end text c o l o u r right parenthesis equals 24 over 48 equals 1 half end cell row cell left parenthesis i i i right parenthesis P left parenthesis a text    end text f a c e text    end text c a r d right parenthesis equals 8 over 48 equals 1 over 6 end cell row cell left parenthesis i v right parenthesis P left parenthesis a text    end text q u e e n right parenthesis equals 2 over 48 equals 1 over 24 end cell end table


 

 

Question 74

In a bag there are 44 identical cards with figure of circle or square on them. There are 24 circles, of which 9 are blue and rest are green and 20 squares of which 11 are blue and rest are green. One card is drawn from the bag at random. Find the probability that it has the figure

  1. square
  2. green colour,
  3. blue circle and
  4. green square.
Solution 74

Question 75

All red face cards are removed from a pack of playing cards. The remaining cards are well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is

  1. a red card
  2. a face card and
  3. a card of clubs.
Solution 75

Question 76

Two customers are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) different days? (iii) consecutive days?

Solution 76

Question 77

Three coins are tossed together. Find the probability of getting :Exactly two heads

Solution 77

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event ofgetting exactly two heads.

A = {HHT, THH, HHT}

n (A) = 3

P(A) =  

Question 78

Three coins are tossed together. Find the probability of getting :At most two heads

Solution 78

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event of getting at most two heads.

A = {TTT, HTT, THT, TTH, HHT, THH, HHT}

n (A) = 7

P(A) =  

Question 79

Three coins are tossed together. Find the probability of getting :At least one head and one tail.

Solution 79

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event of getting at least one head and one tail.

A = {HHT, HTH, THH, HTT, THT, TTH }

n (A) = 6

P(A) =  

Question 80

Three coins are tossed together. Find the probability of getting :No tails

Solution 80

Sample space when three coins are tossed together is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

Let A be the event ofgetting no tails.

A = {HHH}

n (A) = 1

P(A) =  

Question 81

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :

The sum of the two numbers that turn up is 8? 

Solution 81

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 8.

A = {(2,6),(3,5),(4,4),(5,3),(6,2)}

n(A) = 5

P(A) =

Question 82

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :

Of obtaining a total of 6? 

Solution 82

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 6.

A = {(1,5),(2,4),(3,3),(4,2),(5,1)}

n(A) = 5

P(A) =  

Question 83

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :Of obtaining a total of 10?

Solution 83

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the sum of two numbers which turn up is 10.

A = {(4,6),(6,4),(5,5)}

n(A) = 3

P(A) =  

Question 84

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :Of obtaining the same number on both dice?

Solution 84

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event of obtaining the same number on both dice.

A = {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}

n(A) = 6

P(A) =  

Question 85

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that : Of obtaining a total more than 9?

Solution 85

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event of obtaining a total more than 9.

A = {(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)}

n(A) = 6

P(A) =  

Question 86

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the sum of the numbers appearing on the top of the dice is 13?

Solution 86

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the sum of the numbers appearing on the top of the dice is 13.

The maximum total of the numbers on the dice is 12.

Hence, n(A) = 0

P(A) =  

Question 87

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the sum of the numbers appearing on the top of the dice is less than or equal to 12?

Solution 87

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the sum of the numbers appearing on the top of the dice is less than or equal to 12.

The maximum total of the numbers on the dice is 12.

Hence, n(A) = 36

P(A) =  

Question 88

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the product of numbers appearing on the top of dice is less than 9.

Solution 88

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the product of numbers appearing on the top of the dice is less than 9.

A = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

 (2,1)(2,2),(2,3),(2,4), (3,1),(3,2),

 (4,1),(4,2),(5,1),(6,1)}

n(A) = 16

 

P(A) =  

Question 89

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that :That the difference of the numbers appearing on the top of the two dice is 2.

Solution 89

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the difference of the numbers appearing on the top of the two dice is 2.

A = {(1,3),(2,4),(3,1),(3,5),(4,2),(4,6),(5,3),(6,4)}

}

n(A) = 8

P(A) =  

Question 90

A black die and a white die are thrown at the same time. Write all the possible outcomes. What is the probability that : That the numbers obtained have a product less than 16.

Solution 90

When a black die and a white die are thrown at the same time, the sample space is given by

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1)(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

n(S) = 36

Let A be the event that the numbers obtained have a product less than 16.

A = {{(1,1),(1,2),(1,3),(1,4),(1,5),

(1,6),(2,1)(2,2),(2,3),(2,4),(2,5),

(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),

(4,1),(4,2),(4,3), (5,1),(5,2),

(5,3),(6,1),(6,2)}}

n(A) = 25

P(A) =  

Chapter 16 - Probability Exercise Ex. 16.2

Question 1

Solution 1

Question 2



Solution 2

Question 3

A target shown in Fig. 16.11 consists of three concentric circles of radii 3, 7 and 9 cm respectively. A dart is thrown and lands on the target. What is the probability that the dart will land on the shaded region?

 



Solution 3

Assume first circle to be the circle with the smallest radius, that is 3. Similarly, second circle to be the circle with radius 7 and third circle to be the circle with radius 9.

Question 4

In Fig. 16.12, points A, B, C and D are the centres of four circles that each have a radius of length one unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region?


Solution 4

Question 5

In the fig., JKLM is a square with sides of length 6 units. Points A and B are the mid-points of sides KL and LM respectively. If a point is selected at random from the interior of the square. What is the probability that the point will be chosen from the interior of JAB?

Solution 5

Question 6

In the fig., a square dart board is shown. The length of a side of the larger square is 1.5 times the length of a side of the smaller square. If a dart is thrown and lands on the larger square. What is the probability that it will land in the interior of the smaller square?

Solution 6

Chapter 16 - Probability Exercise 16.35

Question 1

If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, then the probability that it is odd, is

begin mathsize 12px style open parentheses straight a close parentheses space 4 over 9 open parentheses straight b close parentheses space 5 over 9 open parentheses straight c close parentheses space 1 over 9 open parentheses straight d close parentheses space 2 over 3 end style

Solution 1

n(E) = total numbers

       = 9

n(0) = odd numbers {1, 3, 5, 7, 9}

       = 5

begin mathsize 12px style straight P left parenthesis straight E right parenthesis space equals space Probability space that space number space is space odd space space space space space space space space space equals space fraction numerator straight n open parentheses 0 close parentheses over denominator straight n open parentheses straight E close parentheses end fraction space space space space space space space space space space equals space 5 over 9 end style

So, the correct option is (b).

Question 2

In Q. No. 1, the probability that the digit is even, is

begin mathsize 12px style open parentheses straight a close parentheses space 4 over 9 open parentheses straight b close parentheses space 5 over 9 open parentheses straight c close parentheses space 1 over 9 open parentheses straight d close parentheses space 2 over 3 end style

Solution 2

n(E) = 9

n(4) = no. is even {2, 4, 6, 8}

       = 4

begin mathsize 12px style straight P left parenthesis straight E right parenthesis space equals space Probability space that space number space is space even space space space space space space space space space equals space fraction numerator straight n open parentheses straight A close parentheses over denominator straight n open parentheses straight E close parentheses end fraction space space space space space space space space space equals space 4 over 9 end style

So, the correct option is (a).

Chapter 16 - Probability Exercise 16.36

Question 1

In Q. No. 1, the probability that the digit is a multiple of 3 is

begin mathsize 12px style open parentheses straight a close parentheses space 1 third open parentheses straight b close parentheses space 2 over 3 open parentheses straight c close parentheses space 1 over 9 open parentheses straight d close parentheses space 2 over 9 end style

Solution 1

n(E) = 9

n(A) = no. is multiple of 3 {3, 6, 9}

       = 3

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight A close parentheses over denominator straight n open parentheses straight E close parentheses end fraction space space space space space space space space space equals space 3 over 9 space space space space space space space space space equals space 1 third end style

So, the correct option is (a).

Question 2

If three coins are tossed simultaneously, then the probability of getting at least two heads, is

begin mathsize 12px style open parentheses straight a close parentheses space 3 over 4 open parentheses straight b close parentheses space 3 over 8 open parentheses straight c close parentheses space 1 half open parentheses straight d close parentheses space 1 fourth end style

Solution 2

3 coins are tossed simultaneously.

Hence sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Event (E) = at least two Heads

              = {HHH, HHT, HTH, THH}

n(s) = 8

n(E) = 4

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space space equals space 4 over 8 space equals space 1 half end style

So, the correct option is (c).

Question 3

In a single throw of a die, the probability of getting a multiple of 3 is

begin mathsize 12px style open parentheses straight a close parentheses space 1 half open parentheses straight b close parentheses space 1 third open parentheses straight c close parentheses space 1 over 6 open parentheses straight d close parentheses space 2 over 3 end style

Solution 3

sample space (s) = {1, 2, 3, 4, 5, 6}

n(s) = 6

Event (E) = getting a multiple of 3

              = {3, 6}

n(E) = 2

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space space equals space 2 over 6 space space space space space space space space space space equals space 1 third end style

So, the correct option is (b).

Question 4

begin mathsize 12px style The space probability space of space guessing space the space correct space answer space to space straight a space certain space test space questions space is space straight x over 12. space If space the space probability space of space not space guessing space the space correct space answer space to space this space question space is space 2 over 3 comma space then space straight x space equals open parentheses straight a close parentheses space 2 open parentheses straight b close parentheses space 3 open parentheses straight c close parentheses space 4 open parentheses straight d close parentheses space 6 end style

Solution 4

begin mathsize 12px style We space know space that comma space for space any space question comma space either space he space gives space the space answer space correctly space or space incorrectly. So comma straight P open parentheses correct space answer close parentheses space plus space straight P open parentheses wrong space answer close parentheses space equals space 1 space space space space space space space space space space...... open parentheses 1 close parentheses It space is space given space that straight P left parenthesis correct space answer right parenthesis space equals space straight x over 12 space space space space space space...... open parentheses 2 close parentheses straight P open parentheses wrong space answer close parentheses space equals space 2 over 3 space space space space space space..... open parentheses 3 close parentheses from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses straight x over 12 space plus space 2 over 3 space equals space 1 straight x over 12 equals space 1 third straight x space equals space 4 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 5

A bag contains three green marbles, four blue marbles, and two orange marbles. If a marble is picked at random, then the probability that it is not an orange marble is

begin mathsize 12px style open parentheses straight a close parentheses space 1 fourth open parentheses straight b close parentheses space 1 third open parentheses straight c close parentheses space 4 over 9 open parentheses straight d close parentheses space 7 over 9 end style

Solution 5

begin mathsize 12px style Total space Marbles space equals 3 space green space plus space 4 space blue space plus space 2 space orange space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 9 Total space orange space marbles space equals space 2 Probability space that space picked space marble space is space an space orange space straight P open parentheses 0 close parentheses space equals space fraction numerator orange space marbles over denominator Total space marble end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 over 9 Probability space that space picked space marble space is space not space orange space equals space 1 space minus space straight P open parentheses 0 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space 2 over 9 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 7 over 9 So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 6

A number is selected at random from the numbers 3, 5, 5, 7, 7, 7, 9, 9, 9, 9 The probability that the selected number is their average is

begin mathsize 12px style open parentheses straight a close parentheses space 1 over 10 open parentheses straight b close parentheses space 3 over 10 open parentheses straight c close parentheses space 7 over 10 open parentheses straight d close parentheses space 9 over 10 end style

Solution 6

begin mathsize 12px style Sample space space space open parentheses straight s close parentheses space equals space left curly bracket 3 comma space 5 comma space 5 comma space 7 comma space 7 comma space 7 comma space 9 comma space 9 comma space 9 comma space 9 right curly bracket straight n open parentheses straight s close parentheses space equals space 10 average space of space these space numbers space equals space fraction numerator 3 space plus space 5 space plus space 5 space plus space 7 space plus space 7 space plus space 7 space plus space 9 space plus space 9 space plus space 9 space plus space 9 over denominator 10 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 70 over 10 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 7 so space we space have space to space find space the space probability space that space the space picked space number space is space 7 space space space space space space space space space space space space space space equals fraction numerator frequency space of space 7 over denominator straight n open parentheses straight s close parentheses end fraction space space space space space space space space space space space space space space equals space 3 over 10 So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 7

The probability of throwing a number greater than 2 with a fair dice is

begin mathsize 12px style open parentheses straight a close parentheses space 3 over 5 open parentheses straight b close parentheses space 2 over 5 open parentheses straight c close parentheses space 2 over 3 open parentheses straight d close parentheses space 1 third end style

Solution 7

Sample space (S) = {1, 2, 3, 4, 5, 6}

n(S) = 6

Event (E) = getting number greater than 2

              = {3, 4, 5, 6}

n(E) = 4

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 4 over 6 space equals space 2 over 3 end style

So, the correct option is (c).

Question 8

A card is accidently dropped from a pack of 52 playing cards. The probability that it is an ace is

begin mathsize 12px style open parentheses straight a close parentheses space 1 fourth open parentheses straight b close parentheses space 1 over 13 open parentheses straight c close parentheses space 1 over 52 open parentheses straight d close parentheses space 12 over 13 end style

Solution 8

n(S) = 52

no. of ace in a pack of 52 cards = 4

n(E) = 4

begin mathsize 12px style Probability space that space dropped space card space is space ace space space straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4 over 52 equals space 1 over 13 end style

So, the correct option is (b).

Question 9

A number is selected from numbers 1 to 25. The probability that it is prime is

begin mathsize 12px style open parentheses straight a close parentheses space 2 over 3 open parentheses straight b close parentheses space 1 over 6 open parentheses straight c close parentheses space 1 third open parentheses straight d close parentheses space 5 over 6 end style

Solution 9

n(S) = 25

Event (E) = prime numbers between 1 to 25

              = {2, 3, 5, 7, 11, 13, 17, 19, 23}

n(E) = 9

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space space equals space 9 over 25 space end style

 

Note: The answer does not match the options in the question.

Question 10

Which of the following cannot be the probability of an event?

begin mathsize 12px style open parentheses straight a close parentheses space 2 over 3 open parentheses straight b close parentheses space minus 1.5 open parentheses straight c close parentheses space 15 percent sign open parentheses straight d close parentheses space 0.7 end style

Solution 10

We know probability P(E) of an event lies between 0 < P(E) < 1     ......(1)

(a), (c), (d) satisfies the (1) but (b) is a negative number. It can't be the probability of an event.

So, the correct option is (b).

Question 11

If P(E) = 0.05, then P(not E) =

(a) - 0.05

(b) 0.5

(c) 0.9

(d) 0.95

Solution 11

We know

P(E) + P(not E) = 1

given P(E) = 0.05

so P(not E) = 1 - 0.05

                 = 0.95

So, the correct option is (d).

Question 12

Which of the following cannot be the probability of occurrence of an event?

(a) 0.2

(b) 0.4

(c) 0.8

(d) 1.6

Solution 12

We know 0 < P(E) < 1

(a), (b), (c) fullfill the condition. But (d) doesn't 

Hence (d) is correct option.

So, the correct option is (d).

Question 13

The probability of a certain event is

(a) 0

(b) 1

(c) 1/2

(d) no existent

Solution 13

An event that is certain to occur is called Certain event.

Probability of certain event is 1.

Ex: If it is Monday, the probability that tomorrow is Tuesday is certain and therefore probability is 1.

So, the correct option is (b).

Chapter 16 - Probability Exercise 16.37

Question 1

The probability of an impossible event is

(a) 0

(b) 1

(c) 1/2

(d) non-existent

Solution 1

Events that are not possible are impossible event.

Probability of impossible event is 0.

So, the correct option is (a).

Question 2

Aarushi sold 100 lottery tickets in which 5 tickets carry prizes. If Priya purchased a ticket, what is the probability of Priya winning a prize?

begin mathsize 12px style open parentheses straight a close parentheses space 19 over 20 open parentheses straight b close parentheses space 1 over 25 open parentheses straight c close parentheses space 1 over 20 open parentheses straight d close parentheses space 17 over 20 end style

Solution 2

begin mathsize 12px style straight n left parenthesis straight S right parenthesis space equals space 100 space space Event space left parenthesis straight E right parenthesis space equals space winnig space straight a space prize space space straight n left parenthesis straight E right parenthesis space equals space There space are space only space 5 space tickets space which space carry space prize. so space straight n left parenthesis straight E right parenthesis space equals space 5 so space probability space of space priya space winnig space straight a space prize space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 5 over 100 equals space 1 over 20 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or 5 ?

begin mathsize 12px style open parentheses straight a close parentheses space 13 over 25 open parentheses straight b close parentheses space 21 over 50 open parentheses straight c close parentheses space 12 over 25 open parentheses straight d close parentheses space 23 over 50 end style

Solution 3

begin mathsize 12px style straight n left parenthesis straight s right parenthesis space equals space 50 If space straight a space number space is space straight a space multiple space of space 5 space equals space left curly bracket 5 comma space 10 comma space 15 comma space 20 comma space 25 comma space 30 comma space 35 comma space 40 comma space 45 comma space 50 right curly bracket straight n left parenthesis straight E 1 right parenthesis space equals space 10 If space straight a space number space is space straight a space multiple space of space 3 space equals space left curly bracket 3 comma space 6 comma space 9 comma space 12 comma space 15 comma space 18 comma space 21 comma space 24 comma space 27 comma space 30 comma space 33 comma space 36 comma space 39 comma space 42 comma space 45 comma space 48 right curly bracket straight n left parenthesis straight E 2 right parenthesis space equals space 16 If space straight a space number space is space straight a space multiple space of space both space 3 space and space 5 space equals space left curly bracket 15 comma space 30 comma space 45 right curly bracket straight n open parentheses straight E 1 space intersection space straight E 2 close parentheses space equals space 3 Probability space that space straight a space number space is space multiple space of space 3 space or space 5 straight P open parentheses straight E 1 space union space straight E 2 close parentheses space equals space straight P open parentheses straight E 1 close parentheses space plus space straight P open parentheses straight E 2 close parentheses space minus space straight P open parentheses straight E 1 space intersection space straight E 2 close parentheses equals space fraction numerator straight n open parentheses straight E 1 close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space plus space fraction numerator straight n open parentheses straight E 2 close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space minus space fraction numerator straight n open parentheses straight E 1 space intersection space straight E 2 close parentheses over denominator straight n open parentheses straight S close parentheses end fraction equals space 10 over 50 space plus space 16 over 50 space minus space 3 over 50 equals space 23 over 50 So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 4

begin mathsize 12px style straight A space month space is space selected space at space random space in space straight a space year. space The space probability space that space it space is space March space or space October comma space is open parentheses straight a close parentheses space 1 over 12 open parentheses straight b close parentheses space 1 over 6 open parentheses straight c close parentheses space 3 over 4 open parentheses straight d close parentheses space None space of space these end style

Solution 4

begin mathsize 12px style Total space month space in space straight a space year comma space straight n open parentheses straight S close parentheses space equals space 12 Month space is space either space march space or space october comma space straight n open parentheses straight E close parentheses space equals space 2 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction equals 2 over 12 equals space 1 over 6 So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

begin mathsize 12px style From space the space letter space of space the space word space

Solution 5

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 6 vowels space in space

Question 6

begin mathsize 12px style straight A space die space is space thrown space once. space The space probability space of space getting space straight a space prime space number space is open parentheses straight a close parentheses space 2 over 3 open parentheses straight b close parentheses space 1 third open parentheses straight c close parentheses space 1 half open parentheses straight d close parentheses space 1 over 6 end style

Solution 6

begin mathsize 12px style If space straight a space dice space is space thrown comma space possible space outcomes space equals space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 close curly brackets straight n open parentheses straight S close parentheses space equals space 6 prime space numbers space equals space open curly brackets 2 comma space 3 comma space 5 close curly brackets straight n open parentheses straight E close parentheses space equals space 3 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space equals space 3 over 6 space equals space 1 half So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

begin mathsize 12px style The space probability space of space getting space an space even space number comma space when space straight a space die space is space thrown space once space is open parentheses straight a close parentheses space 1 half open parentheses straight b close parentheses space 1 third open parentheses straight c close parentheses space 1 over 6 open parentheses straight d close parentheses space 5 over 6 end style

Solution 7

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 6 Even space number space equals space open curly brackets 2 comma space 4 comma space 6 close curly brackets straight n open parentheses straight E close parentheses space equals space 3 straight P open parentheses straight E close parentheses space equals space 3 over 6 space equals space 1 half So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 8

begin mathsize 12px style straight A space box space contains space 90 space discs comma space numbered space from space 1 space to space 90. space If space one space disc space is space drawn space at space random space from space the space box comma space the space probability space that space it space bears space straight a space prime space number space less space than space 23 comma space is open parentheses straight a close parentheses space 7 over 90 open parentheses straight b close parentheses space 10 over 90 open parentheses straight c close parentheses space 4 over 45 open parentheses straight d close parentheses space 9 over 89 end style

Solution 8

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 90 prime space numbers space less space than space 23 space equals space left curly bracket 2 comma space 3 comma space 5 comma space 7 comma space 11 comma space 13 comma space 17 comma space 19 right curly bracket straight n open parentheses straight E close parentheses space equals space 8 straight P open parentheses straight E close parentheses space equals space 8 over 90 space space space space space space space space space equals space 4 over 45 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 9

begin mathsize 12px style The space probability space that space straight a space number space selected space at space random space from space the space number space 1 comma space 2 comma space 3 comma space.... comma space 15 space is space straight a space mutiple space of space 4 comma space is open parentheses straight a close parentheses space 4 over 15 open parentheses straight b close parentheses space 2 over 15 open parentheses straight c close parentheses space 1 fifth open parentheses straight d close parentheses space 1 third end style

Solution 9

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 15 selected space number space is space multiple space of space 4 space equals space open curly brackets 4 comma space 8 comma space 12 close curly brackets straight n open parentheses straight E close parentheses space equals space 3 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space equals space 3 over 15 equals space 1 fifth space So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 10

begin mathsize 12px style Two space different space coins space are space tossed space simultaneously. space The space probability space of space getting space at space least space one space head space is open parentheses straight a close parentheses space 1 fourth open parentheses straight b close parentheses space 1 over 8 open parentheses straight c close parentheses space 3 over 4 open parentheses straight d close parentheses space 7 over 8 end style

Solution 10

begin mathsize 12px style sample space space space open parentheses straight S close parentheses space equals space open curly brackets HH comma space HT comma space TH comma space TT close curly brackets straight n open parentheses straight S close parentheses space equals space 4 sample space having space at space least space one space head space equals space open curly brackets HH comma space HT comma space TH close curly brackets straight n open parentheses straight E close parentheses space equals space 3 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space equals space 3 over 4 space So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 11

begin mathsize 12px style If space two space different space dice space are space rolled space together comma space the space probability space of space getting space an space even space number space on space both space dice comma space is open parentheses straight a close parentheses space 1 over 36 open parentheses straight b close parentheses space 1 half open parentheses straight c close parentheses space 1 over 6 open parentheses straight d close parentheses space 1 fourth end style

Solution 11

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 36 Event space open parentheses straight E close parentheses space equals space open curly brackets open parentheses 2 comma space 2 close parentheses comma space open parentheses 2 comma space 4 close parentheses comma space open parentheses 2 comma space 6 close parentheses comma space open parentheses 4 comma space 2 close parentheses comma space open parentheses 4 comma space 4 close parentheses comma space open parentheses 4 comma space 6 close parentheses comma space open parentheses 6 comma space 2 close parentheses comma space open parentheses 6 comma space 4 close parentheses comma space open parentheses 6 comma 6 close parentheses close curly brackets straight n open parentheses straight E close parentheses space equals space 9 straight P left parenthesis straight E right parenthesis space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 9 over 36 equals space 1 fourth So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 12

begin mathsize 12px style straight A space number space is space selected space at space random space from space the space numbers space 1 space to space 30. space The space probability space that space it space is space straight a space prime space number space is open parentheses straight a close parentheses space 2 over 3 open parentheses straight b close parentheses space 1 over 6 open parentheses straight c close parentheses space 1 third open parentheses straight d close parentheses space 11 over 30 end style

Solution 12

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 30 sample space of space prime space number space equals space open curly brackets 2 comma space 3 comma space 5 comma space 7 comma space 11 comma space 13 comma space 17 comma space 19 comma space 23 comma space 29 close curly brackets straight n open parentheses straight E close parentheses space equals space 10 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 10 over 30 space equals space 1 third So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 16 - Probability Exercise 16.38

Question 1

begin mathsize 12px style straight A space card space is space drawn space at space random space from space straight a space pack space of space 52 space cards. space The space probability space that space the space drawn space card space is space not space an space ace space is open parentheses straight a close parentheses space 1 over 13 open parentheses straight b close parentheses space 9 over 13 open parentheses straight c close parentheses space 4 over 13 open parentheses straight d close parentheses space 12 over 13 end style

Solution 1

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 52 There space are space 4 space ace space in space straight a space pack space of space 52 space cards Hence space straight n open parentheses straight E close parentheses space equals space 4 Probability space that space drawn space card space is space ace space comma space straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction equals space 4 over 52 space equals space 1 over 13 Probability space that space drawn space card space is space not space ace space equals space 1 space minus space card space is space ace space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space straight P open parentheses straight E close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space 1 over 13 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 12 over 13 So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 2

A number x is chosen at random from the numbers -3, -2, -1, 0, 1, 2, 3 the probability that |x| < 2 is

begin mathsize 12px style open parentheses straight a close parentheses space 5 over 7 open parentheses straight b close parentheses space 2 over 7 open parentheses straight c close parentheses space 3 over 7 open parentheses straight d close parentheses space 1 over 7 end style

Solution 2

Sample space (s) = {-3, -2, -1, 0, 1, 2, 3}

n(s) = 7

Event (E) = |x| < 2

             = {-1, 0, 1}

n(E) = 3

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight s close parentheses end fraction space equals space 3 over 7 end style

So, the correct option is (c).

Question 3

If a number x is chosen from the numbers 1, 2, 3, and a number y is selected from the number 1, 4, 9. Then, P(xy < 9)

begin mathsize 12px style open parentheses straight a close parentheses space 7 over 9 open parentheses straight b close parentheses space 5 over 9 open parentheses straight c close parentheses space 2 over 3 open parentheses straight d close parentheses space 1 over 9 end style

Solution 3

begin mathsize 12px style straight x space element of space open curly brackets 1 comma space 2 comma space 3 close curly brackets straight y space element of space open curly brackets 1 comma space 4 comma space 9 close curly brackets Now space set space xy space element of space left curly bracket 1 space cross times space 1 comma space 1 space cross times space 4 comma space 1 space cross times space 9 comma space 2 space cross times space 1 comma space 2 space cross times space 4 comma space 2 space cross times space 9 comma space 3 space cross times space 1 comma space 3 space cross times space 4 comma space 3 space cross times space 9 right curly bracket space space space space space space space space space space space space space space space space space space xy space element of space open curly brackets 1 comma space 4 comma space 9 comma space 2 comma space 8 comma space 18 comma space 3 comma space 12 comma space 27 close curly brackets sample space space space straight n open parentheses straight S close parentheses comma space xy space element of space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 8 comma space 9 comma space 12 comma space 18 comma space 27 close curly brackets straight n open parentheses straight S close parentheses space equals space 9 Event space open parentheses straight E close parentheses space equals space xy space less than space 9 space space space space space space space space space space space space space space space space space space space equals open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 8 close curly brackets straight n open parentheses straight E close parentheses space equals space 5 straight P open parentheses xy space less than space straight a close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 5 over 9 So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 4

The probability that a non-leap year has 53 Sundays, is

begin mathsize 12px style open parentheses straight a close parentheses space 2 over 7 open parentheses straight b close parentheses space 5 over 7 open parentheses straight c close parentheses space 6 over 7 open parentheses straight d close parentheses space 1 over 7 end style

Solution 4

There are 365 days in a non-leap year.

52 complete weeks and 1 spare day.

so This day can be any out of 7 day of week.

Hence n(s) = 7

Now, year already have 52 Sundays. so for a total of 53 Sundays in a calendar year, this spare day must be a Sunday.

Hence n(E) = 1

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space probability space that space non space leap space year space has space 53 space sundays space space space space space space space space space equals space 1 over 7 end style

So, the correct option is (d).

Question 5

In a single throw of a pair of dice, the probability of getting the sum a perfect sqaure is

begin mathsize 12px style open parentheses straight a close parentheses space 1 over 18 open parentheses straight b close parentheses space 7 over 36 open parentheses straight c close parentheses space 1 over 6 open parentheses straight d close parentheses space 2 over 9 end style

Solution 5

We know on throwing a pair of die there are a total of 36 possible outcomes.

n(S) = sum is a perfect square

        = {(1, 3), (2, 2), (3, 1), (3, 6), (4, 5), (5, 4), (6, 3)}

n(E) = 7

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 7 over 36 end style

So, the correct option is (b).

 

Question 6

What is the probability that a non-leap year has 53 Sundays ?

begin mathsize 12px style open parentheses straight a close parentheses space 6 over 7 open parentheses straight b close parentheses space 1 over 7 open parentheses straight c close parentheses space 5 over 7 open parentheses straight d close parentheses space None space of space these end style

Solution 6

There are 365 days in a non-leap year.

52 complete weeks and 1 spare day.

So this day can be any out of 7 day of a week.

Hence n(s) = 7

Now, a non-leap year already has 52 Sundays. So for a total of 53 Sundays in a calendar year, this spare day must be a Sunday.

Hence n(E) = 1

begin mathsize 12px style straight P open parentheses straight E close parentheses space equals space probability space that space non space leap space year space has space 53 space sundays space space space space space space space space space equals space 1 over 7 end style

So, the correct option is (b).

Question 7

Error converting from MathML to accessible text.

Solution 7

Error converting from MathML to accessible text.

Question 8

Two dice are rolled simultaneously. The probability that they show different faces is

begin mathsize 12px style open parentheses straight a close parentheses space 2 over 3 open parentheses straight b close parentheses space 1 over 6 open parentheses straight c close parentheses space 1 third open parentheses straight d close parentheses space 5 over 6 end style

Solution 8

begin mathsize 12px style If space two space dices space are space rolled space simultaneously space there space are space 36 space possible space outcomes. straight n open parentheses straight S close parentheses space equals space 36 Event space open parentheses straight E close parentheses space equals space faces space show space same space number space space space space space space space space space space space space space space space space space space space space equals space left curly bracket open parentheses 1 comma space 1 close parentheses comma space open parentheses 2 comma space 2 close parentheses comma space open parentheses 3 comma space 3 close parentheses comma space open parentheses 4 comma space 4 close parentheses comma space open parentheses 5 comma space 5 close parentheses comma space open parentheses 6 comma space 6 close parentheses right curly bracket straight n open parentheses straight E close parentheses space equals space 6 Probability space that space faces space show space same space number space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction equals space 6 over 36 space equals space 1 over 6 Probability space that space faces space show space different equals space 1 space minus space straight P open parentheses straight E close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space 1 over 6 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 5 over 6 space space space space space space space space space space space space space space space space space space space So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 9

What is the probability that a leap year has 52 Mondays?

begin mathsize 12px style open parentheses straight a close parentheses space 2 over 7 open parentheses straight b close parentheses space 4 over 7 open parentheses straight c close parentheses space 5 over 7 open parentheses straight d close parentheses space 6 over 7 end style

Solution 9

begin mathsize 12px style straight A space leap space year space has space 366 space days space in space which space 52 space weeks space and space 2 space spare space days. So space for space straight a space leap space year space to space have space 52 space Mondays comma space non space of space the space spare space days space can space be space straight a space Monday. Possible space combinations space for space these space two space days sample space space space open parentheses straight S close parentheses space equals space open curly brackets open parentheses Monday comma space Tuesday close parentheses comma space open parentheses Tuesday comma space Wednesday close parentheses comma space open parentheses Wednesday comma space Thursday close parentheses comma space open parentheses Thursday comma space Friday close parentheses comma space open parentheses Friday comma space Saturday close parentheses comma space open parentheses Saturday comma space Sunday close parentheses comma space open parentheses Sunday comma space Monday close parentheses close curly brackets straight n open parentheses straight S close parentheses space equals space 7 Event space that space none space of space the space combinations space have space monday straight n open parentheses straight E close parentheses space equals space 5 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 5 over 7 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 10

begin mathsize 12px style If space straight a space two space digit space number space is space chosen space at space random comma space then space the space probability space that space the space number space chosen space is space straight a space multiple space of space 3 comma space is open parentheses straight a close parentheses space 3 over 10 open parentheses straight b close parentheses space 29 over 100 open parentheses straight c close parentheses space 1 third open parentheses straight d close parentheses space 7 over 25 end style

Solution 10

begin mathsize 12px style two space digit space number space is space from space 10 space to space 99 Hence space straight n open parentheses straight S close parentheses space equals space 90 If space number space choosen space is space straight a space multiple space of space 3 space then space it space starts space from space 12 space to space 99 space at space an space interval space of space 3. Hence space order space is space 12 comma space 15 comma space 18 comma space........... space 99 It space is space an space AP space with space straight a space equals space 12 space and space straight d equals space 3 comma space last space term space 99 straight a space plus space open parentheses straight r space minus space 1 close parentheses straight d space equals space 99 12 space plus space open parentheses straight r space minus space 1 close parentheses 3 space equals space 99 open parentheses straight r space minus space 1 close parentheses space 3 space equals space 87 open parentheses straight r space minus space 1 close parentheses space equals space 29 straight r space equals space 30 Hence space straight n open parentheses straight E close parentheses space equals space 30 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 30 over 90 space equals space 1 third So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 11

begin mathsize 12px style Two space dice space are space thrown space together. space The space probability space of space getting space the space same space number space on space both space dice space is open parentheses straight a close parentheses space 1 half open parentheses straight b close parentheses space 1 third open parentheses straight c close parentheses space 1 over 6 open parentheses straight d close parentheses space 1 over 12 end style

Solution 11

begin mathsize 12px style When space we space throw space two space dice space together straight n open parentheses straight S close parentheses space equals space 36 Event space that space both space dice space have space same space number space is open curly brackets open parentheses 1 comma space 1 close parentheses comma space open parentheses 2 comma space 2 close parentheses comma space open parentheses 3 comma space 3 close parentheses comma space open parentheses 4 comma space 4 close parentheses comma space open parentheses 5 comma space 5 close parentheses comma space open parentheses 6 comma space 6 close parentheses close curly brackets straight n open parentheses straight E close parentheses space equals space 6 straight P open parentheses straight E close parentheses space equals space 6 over 36 space equals space 1 over 6 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 16 - Probability Exercise 16.39

Question 1

begin mathsize 12px style In space straight a space family space of space 3 space children comma space the space probability space of space having space at space least space one space boy space is open parentheses straight a close parentheses space 7 over 8 open parentheses straight b close parentheses space 1 over 8 open parentheses straight c close parentheses space 5 over 8 open parentheses straight d close parentheses space 3 over 4 end style

Solution 1

begin mathsize 12px style straight B space minus space Boy straight G space minus space Girl Possible space combinations space equals space open curly brackets BBB comma space BBG comma space BGB comma space GBB comma space BGG comma space GBG comma space GGB comma space GGG close curly brackets straight n open parentheses straight S close parentheses space equals space 8 probability space having space at space least space one space boy space equals space 1 space minus space probability space of space having space no space boys cases space in space which space no space boys comma space straight n open parentheses straight E close parentheses space equals space 1 Probability space of space having space no space boys space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space equals space 1 over 8 probability space having space at space least space one space boy space equals space 1 space minus space 1 over 8 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 7 over 8 So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 12px style straight A space bag space contains space cards space numbered space from space 1 space to space 25. space straight A space card space is space drawn space at space random space from space the space bag. space The space probability space that space the space number space on space this space card space is space divisible space by space both space 2 space and space 3 space is open parentheses straight a close parentheses space 1 fifth open parentheses straight b close parentheses space 3 over 25 open parentheses straight c close parentheses space 4 over 25 open parentheses straight d close parentheses space 2 over 25 space end style

Solution 2

begin mathsize 12px style straight n open parentheses straight S close parentheses space equals space 25 numbers space which space are space divisible space by space both space 2 space and space 3 space equals space open curly brackets 6 comma space 12 comma space 18 comma space 24 close curly brackets straight n open parentheses straight E close parentheses space equals space 4 straight P open parentheses straight E close parentheses space equals space fraction numerator straight n open parentheses straight E close parentheses over denominator straight n open parentheses straight S close parentheses end fraction space space space space space space space space space equals space 4 over 25 So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

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