**Question 1.Write sample space ‘S’ and number of sample points n(S) for each of the following experiments. Also write events A, B, C in the set form and write n(A), n(B), n(C).**

i. One die is rolled,

Event A: Even number on the upper face.

Event B: Odd number on the upper face.

Event C: Prime number on the upper face.

ii. Two dice are rolled simultaneously,

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

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

Event C: The same digit on both the upper faces.

iii. Three coins are tossed simultaneously.

Condition for event A: To get at least two heads.

Condition for event B: To get no head.

Condition for event C: To get head on the second coin.

iv. Two digit numbers are formed using digits 0, 1, 2, 3, 4, 5 without repetition of the digits.

Condition for event A: The number formed is even.

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

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

v. From three men and two women, environment committee of two persons is to be formed.

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

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

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

vi. One coin and one die are thrown simultaneously.

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

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

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

Solution:

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

∴ n(S) = 6

Condition for event A: Even number on the upper face.

∴ A = {2,4,6}

∴ n(A) = 3

Condition for event B: Odd number on the upper face.

∴ B = {1, 3, 5}

∴ n(B) = 3

Condition for event C: Prime number on the upper face.

∴ C = {2, 3, 5}

∴ n(C) = 3

ii. Sample space,

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

Condition for event A: The sum of the digits on the upper faces is a multiple of 6.

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

∴ n(A) = 6

Condition for 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

Condition for 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

iii. Sample space,

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

∴ n(S) = 8

Condition for event A: To get at least two heads.

∴ A = {HHT, HTH, THH, HHH}

∴ n(A) = 4

Condition for event B: To get no head.

∴ B = {TTT}

∴ n(B) = 1

Condition for event C: To get head on the second coin.

∴ C = {HHH, HHT, THH, THT}

∴ n(C) = 4

iv. 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

v. Let the three men be M_{1}, M_{2}, M_{3} and the two women be W_{1}, W_{2}.

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

∴ Sample space,

S = {M_{1}M_{2}, M_{1}M_{3}, M_{1}W_{1}, M_{1}W_{2}, M_{2}M_{3}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{1}, M_{3}W_{2}, W_{1}W_{2}}

∴ n(S) = 10

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

∴ A = {M_{1}W_{1}, M_{1}W_{2}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{1}, M_{3}W_{2}, W_{1}W_{2}}

∴ n(A) = 7

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

∴ B = {M_{1}W_{1}, M_{1}W_{2}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{2}, M_{3}W_{2}}

∴ n(B) = 6

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

∴ C = {M_{1}M_{2}, M_{1}M_{3}, M_{2}M_{3}}

∴ n(C) = 3

vi. Sample space,

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