RD SHARMA Solutions for Class 10 Maths Chapter 16 - Probability

Chapter 16 - Probability Exercise Ex. 16.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 6(xix)

  

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19(i)

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

Solution 19(ii)

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

Solution 20

Solution 21

Solution 22

Solution 23

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.

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

(i) 1

(ii) 0

(iii) 0 and 1

(iv) equal

(v) 1

(vi) 1

Solution 42

Solution 43

Solution 44

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

Solution 45

 

 

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

 

 

 

Solution 46

Solution 47

 

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

 

 

 

Solution 48

Solution 49

Solution 50

 

 
 
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
 
 
Solution 50(xvi)

  

Solution 50(xvii)

  

Solution 51

Solution 52

Solution 53



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

(x)

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) =  

 

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61

Solution 62

Solution 63

Solution 64

Solution 65

 

 

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

 

 

Solution 66

Solution 67

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

Solution 68

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

Solution 69

 

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


 

 

Solution 70

Solution 71

Solution 72

Solution 3(i)

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) =   

Solution 3(ii)

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) =   

Solution 3(iii)

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) =   

Solution 3(iv)

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) =   

Chapter 16 - Probability Exercise Ex. 16.2

Solution 1

Solution 2

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.

Solution 4

Solution 5

Solution 6

Chapter 16 - Probability Exercise 16.35

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).

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

Solution 3

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).

Solution 4

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).

Solution 5

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).

Solution 6

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

Solution 7

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

Solution 8

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

Solution 9

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).

Solution 10

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).

Solution 11

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.

Solution 12

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).

Solution 13

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).

Solution 14

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).

Solution 15

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

Solution 16

Events that are not possible are impossible event.

Probability of impossible event is 0.

So, the correct option is (a).

Solution 17

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

Solution 18

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

Solution 19

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

Solution 20

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

Solution 21

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

Solution 22

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

Solution 23

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

Solution 24

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

Solution 25

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

Solution 26

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

Solution 27

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

Solution 28

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

Solution 29

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).

Solution 30

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

Solution 31

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).

Solution 32

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).

 

Solution 33

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).

Solution 34

Error converting from MathML to accessible text.

Solution 35

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

Solution 36

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

Solution 37

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

Solution 38

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

Solution 39

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

Solution 40

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