Three coins are tossed. Describe (i)Two events which are mutually exclusive. (ii)Three events which are mutually exclusive and exhaustive.(iii)Two events, which are not mutually exclusive.(iv)Two events which are mutually exclusive but not exhaustive.(v)Three events which are mutually exclusive but not exhaustive.

Question

For two events A and B, if \(P(A) = P(A/B) = \frac{1}{4}\) and \(P(B/A) = \frac{1}{2}\), then which of the following is not true?

Answer 1

Sample Space:

When three coins are tossed, the sample space is:

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

(i) Two events which are mutually exclusive

Let,
A = Getting all heads = { HHH }
B = Getting all tails = { TTT }

A ∩ B = ∅, hence A and B are mutually exclusive.

(ii) Three events which are mutually exclusive and exhaustive

Let,
A = Getting no head = { TTT }
B = Getting exactly one head = { HTT, THT, TTH }
C = Getting two or more heads = { HHT, HTH, THH, HHH }

A ∪ B ∪ C = S and A, B, C have no common outcomes.
Hence, they are mutually exclusive and exhaustive.

(iii) Two events which are not mutually exclusive

Let,
A = Getting at least one head = { HHH, HHT, HTH, HTT, THH, THT, TTH }
B = Getting exactly two heads = { HHT, HTH, THH }

A ∩ B ≠ ∅, hence A and B are not mutually exclusive.

(iv) Two events which are mutually exclusive but not exhaustive

Let,
A = Getting all heads = { HHH }
B = Getting exactly one head = { HTT, THT, TTH }

A ∩ B = ∅, but A ∪ B ≠ S.
Hence, they are mutually exclusive but not exhaustive.

(v) Three events which are mutually exclusive but not exhaustive

Let,
A = Getting all heads = { HHH }
B = Getting exactly one head = { HTT, THT, TTH }
C = Getting all tails = { TTT }

A, B, and C are mutually exclusive, but
A ∪ B ∪ C ≠ S.

Hence, they are mutually exclusive but not exhaustive.

Solution and Explanation