Question

Suppose A = {1,2,};B = {2,3,4,6}, C = {5,7} and S = {1,2,3,4,5,6,7,9}

a) What is P(A), what is P(B), what is P(A ∪ B)

b) Compute P(A ∩ B) without computing A ∩ C; compute P(B) without

̄

(A ∪ B) and (A ∪ C) mutually exclusive?

̄

computing B.

c) Are A, B and C collectively exhaustive? Are they mutually exclusive?

Answer 1

Solution:

a) Probability of an event A is given by,

  

Where, n(A) is number of elements in A and n(S) is number of elements in sample space S.

We have, n(A) = 2 and n(S) = 8

Hence, P(A) = 2/8 = 1/4

Probability of an event B is given by,

  

Where, n(B) is number of elements in B and n(S) is number of elements in sample space S.

We have, n(B) = 4 and n(S) = 8

Hence, P(B) = 4/8 = 1/2

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Where,

n(A ∩ B) is number of elements which are available in both of A and B.

Since, only one element is available in both of A and B, therefore, n(A ∩ B) = 1.

n(S) = 8

P(A ∩ B) = 1/8

and we have, P(A) = 1/4, P(B) = 1/2

b)

n(A ∩ B) is number of elements which are available in both of A and B.

Since, only one element is available in both of A and B, therefore, n(A ∩ B) = 1.

n(S) = 8

P(A ∩ B) = 1/8

Already we have computed P(B) = 1/2 in part (a).

Two sets are said to be mutually exclusive if no element is common for both the sets.

A = {1,2,} B = {2,3,4,6} and C = {5,7}

(A ∪ B) = {1,2,3,4,6}

(A ∪ C) = {1,2,5,7}

Since, the two elements are common in both of (A ∪ B) and (A ∪ C), therefore (A ∪ B) and (A ∪ C) are not mutually exclusive.

Howet we don't need to obtain (A ∪ B) and (A ∪ C) to tell whether these two are mutually exclusive or not. Since, always the elements of A will be common in both of (A ∪ B) and (A ∪ C), therefore these two sets are not mutually exclusive.

c) The three sets A, B and C are said to be collectively exhaustive if,

(A ∪ B ∪ C) = S

We have, A = {1,2}; B = {2,3,4,6}; C = {5,7}; S = {1,2,3,4,5,6,7,9}

Hence, (A ∪ B ∪ C) = {1,2,3,4,5,6,7}

Since, therefore A, B and C are not collectively exhaustive.

Three sets A, B and C are said to be mutually exclusive if no elements is common in all of A, B and C.

Since, there is no element which is common in all of A, B and C, therefore A, B and C are mutually exclusive.

Please rate the answer. Thank you.