Class 10 MAHARASHTRA STATE TEXTBOOK BUREAU Solutions Maths Chapter 5 - Probability

As there 8 sites of Maharashtra are given.

∴ There are 8 possibilities of visiting a site out of 8 sites.

There are 7 days in a week.

∴ There are 7 possibilities of selecting any day of a week.

There are 52 cards in a pack of cards.

∴ There are 52 possibilities of selecting a card from the pack of 52 cards.

There are 11 numbers from 10 to 20.

So, there will be 11 cards number from 10 to 20.

∴ There are 11 possibilities of selecting one card from group of given cards.

When one coin and one die are thrown simultaneously, sample space is given by:

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

∴ n(S) =12

Here, sample space will be:

S = {23, 25, 32, 35, 52, 53}

∴ n(S) = 6

There are total 6 colours shown on the disc.

∴ Sample space (S) = {Red, Orange, Yellow, Blue, Green, Purple}

∴ n(S) = 6

∴ The arrow may stop on any one of the 6 colours.

The multiples of 5 are 5, 10, 15, 20, …

∴ The dates with multiples of 5 are 5, 10, 15, 20, 25, 30.

∴ S = {Tuesday, Sunday, Friday, Wednesday, Monday, Saturday}

∴ The days on which the date is a multiple of 5 are Sunday, Monday, Tuesday, Wednesday, Friday and Saturday.

(a) Committee of 2 boys =

(b) Committee of 2 girls =

(c) Committee of one boy and one girl =

∴ Sample space = {B₁B₂, G₁G₂, B₁G₁, B₁G₂, B₂G₁, B₂G₂}

When a die is rolled:

Sample space will be S = {1, 2, 3, 4, 5, 6}

n(S) = 6

Event A : Even number on the upper face

∴ A = {2, 4, 6}

∴ n(A) = 3

Event B : Odd number on the upper face

∴ B = {1, 3, 5}

∴ n(B) = 3

Event C : Prime number on the upper face

∴ C = {2, 3, 5}

∴ n(C) = 3

When two dice are rolled simultaneously:

Sample space will be 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

Event A : The sum of the digits on upper faces is a multiple of 6

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

∴ n(A) = 6

Event B : The sum of the digits on the upper faces is minimum 10

∴ B = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}

∴ n(B) = 6

Event C : The same digit on both the upper faces

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

∴ n(C) = 6

When three coins are tossed simultaneously:

Sample space will be S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }

∴ n(S) = 8

Event A : To get at least two heads

∴ A = {HHT, HTH, THH, HHH}

∴ n(A) = 4

Event B : To get no head

∴ B = {TTT}

∴ n(B) = 1

Event C : To get head on the second coin

∴ C = {HHH, HHT, THH, THT}

∴ n(C) = 4

When two digit numbers are formed using digits 0, 1, 2, 3, 4, 5 without repetition of the digits:

Sample space (S) = {10, 12, 13, 14, 15, 20, 21, 23, 24, 25, 30, 31, 32, 34, 35,

40, 41, 42, 43, 45, 50, 51, 52, 53, 54}

∴ n(S) = 25

Condition for event A: The number formed is even

∴ A = {10, 12, 14, 20, 24, 30, 32, 34, 40, 42, 50, 52, 54}

∴ n(A) = 13

Condition for event B: The number formed is divisible by 3.

∴ B = {12, 15, 21, 24, 30, 42, 45, 51, 54}

∴ n(B) = 9

Condition for event C: The number formed is greater than 50.

∴ C = {51,52, 53,54}

∴ n(C) = 4

Let the three men be M₁, M₂, M₃ and the two women be W₁, W₂.

Out of these men and women, an environment committee of two persons is to be formed.

∴ Sample space will be:

S = {M₁M₂, M₁M₃, M₁W₁, M₁W₂, M₂M₃, M₂W₁, M₂W₂, M₃W₁, M₃W₂, W₁W₂}

∴ n(S) = 10

Condition for event A: There must be at least one woman member

∴ A = {M₁W₁, M₁W₂, M₂W₁, M₂W₂, M₃W₁, M₃W₂, W₁W₂}

∴ n(A) = 7

Condition for event B: One man, one woman committee to be formed

∴ B = {M₁W₁, M₁W₂, M₂W₁, M₂W₂, M₃W₂, M₃W₂}

∴ n(B) = 6

Condition for event C: There should not be a woman member

∴ C = {M₁M₂, M₁M₃, M₂M₃}

∴ n(C) = 3

When one coin and one die are thrown simultaneously:

Sample space will be:

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

∴ n(S) = 12

Condition for event A: To get head and an odd number

∴ A = {(H, 1), (H, 3), (H, 5)}

∴ n(A) = 3

Condition for event B: To get a head or tail and an even number

∴ B = {(H, 2), (H, 4), (H, 6), (T, 2), (T, 4), (T, 6)}

∴ n(B) = 6

Condition for event C: Number on the upper face is greater than 7 and tail on the coin.

The greatest number on the upper face of a die is 6.

∴ Event C is an impossible event.

∴ C = { }

∴ n(C) = 0

When two coins are tossed, the sample space will be:

S = {HH, HT, TH, TT}

(1)

Let A be an event of getting at least one head

∴ A = {HH, HT, TH}

∴ n(A) = 3

(2)

Let B be an event of getting no head

∴ B = {TT}

∴ n(B) = 1

When two dice are rolled simultaneously, the sample space will be

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)}

(1)

Let A be an event that the sum of the digits on the upper faces is at least 10.

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

∴ n(A) = 6

(2)

Let B be the event that the sum of the digits on the upper faces is 33.

Now, the sum of the digits on the upper faces can be maximum 12.

∴ Event B is an impossible event.

∴ B = { }

∴ n(B) = 0

(3)

Let C be an event that the digit on the first die is greater than the digit on second die.

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

∴ n(C) = 15

Here, sample space S = {1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,14, 15}

∴ n(S) = 15

(1)

Let A be the event that the ticket drawn shows an even number.

∴ A = {2, 4, 6, 8, 10, 12, 14}

∴ n(A) = 7

(2)

Let B be the event that the ticket drawn shows a number which is a multiple of 5.

∴ B = {5, 10, 15}

∴ n(B) = 3

Here, sample space S = {23, 25, 27, 29, 32, 35, 37, 39, 52, 53, 57, 59, 72, 73, 75, 79, 92, 93, 95, 97}

∴ n(S) = 20

(1)

Let A be the event that the number formed is an odd number.

∴ A = {23, 25, 27, 29, 35, 37, 39, 53, 57, 59, 73, 75,79,93,95,97}

∴ n(A) = 16

(2)

Let B be the event that the number formed is a multiple of 5.

∴ B = {25,35,75,95}

∴ n(B) = 4

There are 52 playing cards in a pack.

∴ n(S) = 52

(1)

Let A be the event that the card drawn is an ace.

∴ n(A) = 4

(2)

Let B be the event that the card drawn is a spade.

∴ n(B) = 13

Since, the probability is always between 0 and 1.

∴ 1.5 cannot represent a probability.

When a die is rolled, the sample space will be:

S = {1, 2, 3, 4, 5, 6}

∴ n(S) = 6

Let A be an event that the number appearing on upper face is less than 3.

∴ A = {1, 2}

∴ n(A) = 2

Here, sample space is S = {1, 2, 3, 4, …, 100}

∴ n(S) = 100

Event A: Getting a prime number

∴ A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}

∴ n(A) = 25

Here, sample space is S = {1, 2, 3, 4, … 40}

∴ n(S) = 40

Let A be an event of that the card drawn bears a number which is a multiple of 5.

∴ A = {5, 10, 15, 20, 25, 30, 35, 40}

∴ n(A) = 8

The probability that the ball is dropped in the basket by John =

The probability that the ball is dropped in the basket by Vasim = 0.83

The probability that the ball is dropped in the basket by Akash =

Since, 0.58 < 0.8 < 0.83

∴ Vasim had the greatest probability of success.

Here,

Total number of players in the team = 6 + 4 + 1 = 11

∴ n(S) = 11

(1)

Let A be an event that the captain selected is a goalee.

Since, there is only one goalee.

∴ n(A) = 1

(2)

Let B be an event that the captain selected is a defender.

Since there 6 defenders.

∴ n(B) = 6

There are 26 cards kept each bearing one English alphabet.

∴ n(S) = 26

Let A be an event that the card drawn is a vowel card.

∴ A = {a, e, i, o, u}

∴ n(A) = 5

Let the 2 red balloon be R₁, R₂

3 blue balloons be B₁, B₂, B₃, and

4 green balloons be G₁, G₂, G₃, G₄.

∴ Sample space is S = {R₁, R₂, B₁, B₂, B₃, G₁, G₂, G₃, G₄}

∴ n(S) = 9

(1)

Let A be an event that Pranali get s red balloon.

∴ A = {R₁, R₂}

∴ n(A) = 2

∴ The probability that Pranali gets a red balloon is

(2)

Let B be an event that Pranali gets a blue balloon.

∴ B = {B₁, B₂, B₃}

∴ n(B) = 3

∴ The probability that Pranali gets a red balloon is

(3)

Let C be an event that Pranali gets a blue balloon.

∴ C = {G₁, G₂, G₃, G₄}

∴ n(C) = 4

∴ The probability that Pranali gets a red balloon is

Let 5 red pens be R₁, R₂, R₃, R₄, R₅.

8 blue pens be B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈

3 green pens be G₁, G₂, G₃.

∴ Sample space is:

S = {R₁, R₂, R₃, R₄, R₅, B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈, G₁, G₂, G₃}

∴ n(S) = 16

Let A be an event that Rutuja picks a blue pen.

∴ A = {B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈}

∴ n(A) = 8

∴ The probability that Rutuja picks a blue pen is

Here, S = {A, B, C, D, E, A}

∴ n(S) = 6

(1)

Let X be the event that 'A' appears on the upper face.

∴ X = {A, A}

∴ n(X) = 2

(2)

Let Y be the event that 'D' appears on the upper face.

∴ X = {D}

∴ n(X) = 1

Here,

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

∴ n(S) = 30

(1)

Let A be the event that the ticket drawn bears an odd number.

∴ A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}

∴ n(A) = 15

(2)

Let B be the event that the ticket drawn bears a complete square number.

∴ B = {1, 4, 9, 16, 25}

∴ n(B) = 5

Area of the rectangular garden = length × breadth

= 77 × 50

= 3850 sq. m

∴ Area of the rectangular garden = 3850 sq. m

Radius of the lake = 14/2 = 7 m

Area of the circular lake

∴ Area of the lake = 154 sq. m

Probability that the towel fell in the lake

∴ The probability that the towel fell in the lake is

Sample space (S) = {1,2, 3, 4, 5, 6, 7, 8}

∴ n(S) = 8

(1)

Let A be an event that the spinning arrow comes to rest at 8.

∴ A = {8}

∴ n(A) = 1

(2)

Let B be an event that the spinning arrow comes to rest at an odd number.

∴ B = {1, 3, 5, 7}

∴ n(B) = 4

(3)

Let C be an event that the spinning arrow comes to rest at a number greater than 2.

∴ C = {3, 4, 5, 6, 7, 8}

∴ n(C) = 6

(4)

Let D be an event that the spinning arrow comes to rest at a number less than 9.

∴ D = {1, 2, 3, 4, 5, 6, 7, 8}

∴ n(D) = 8

Sample space (S) = {0, 1, 2, 3, 4, 5}

∴ n(S) = 6

(1)

Let A be an event that the card drawn shows a natural number.

∴ A = {1, 2, 3, 4, 5}

∴ n(A) = 5

(2)

Let B be an event that the card drawn shows a number less than 1.

∴ B = {0}

∴ n(B) = 1

(3)

Let C be an event that the card drawn shows a whole number.

∴ C = {0, 1, 2, 3, 4, 5}

∴ n(C) = 6

(4)

Let D be an event that the card drawn shows a number greater than 5.

Here, the greatest number is 5.

∴ D is an impossible event

∴ D = { }

∴ n(D) = 0

∴ P(D) = 0

Let the three red balls be R₁, R₂, R₃, three white balls be W₁, W₂, W₃ and three green balls be G₁, G₂, G₃.

∴ Sample space,

S = { R₁, R₂, R₃, W₁, W₂, W₃, G₁, G₂, G₃}

∴ n(S) = 9

(1)

Let A be the event that the ball drawn is red.

∴ A = {R₁, R₂, R₃}

∴ n(A) = 3

(2)

Let B be the event that the ball drawn is not red.

B = {W₁, W₂, W₃, G₁, G₂, G₃}

∴ n(B) = 6

(3)

Let C be the event that the ball drawn is red or white.

∴ C = {R₁, R₂, R₃, W₁, W₂, W₃}

∴ n(C) = 6

Here, sample space = {m, a, t, h, e, m, a, t, i, c, s}

∴ n(S) = 11

Let A be the event that the card drawn bears the letter 'm'

∴ A = {m, m}

∴ n(A) = 2

∴ The probability that a card drawn bears letter 'm' is

Total number of students in the school = 200

∴ n(S) = 200

Number of students who like Kabaddi = 135

∴ Number of students who do not like Kabaddi = 200 - 135 = 65

Let A be the event that the student selected does not like Kabaddi.

∴ n(A) = 65

∴ The probability that the student selected doesn't like kabaddi is

Sample space (S) = {10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34,

40, 41, 42, 43, 44}

∴ n(S) = 20

(1)

Let A be the event that the number so formed is a prime number.

∴ A = {11,13,23,31,41,43}

∴ n(A) = 6

(2)

Let B be the event that the number so formed is a multiple of 4.

∴ B = {12,20,24,32,40,44}

∴ n(B) = 6

(3)

Let C be the event that the number so formed is a multiple of 4.

∴ C = {11, 22, 33, 44}

∴ n(C) = 4

S = {(0, 0), (0, 1), (0, 2), (1,0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (4, 2), (5, 0), (5, 1), (5, 2)}

∴ n(S) = 36

Let A be the event that the product of digits on the upper face is zero.

∴ A = {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 0), (2, 0), (3, 0), (4, 0), (5,0)}

∴ n(A) = 11

∴ The probability that the product of the digits on the upper face is zero is