From a group of 2 boys and 3 girls, two children are selected at random. Describe the events :

(i) A = event that both the selected children are girls

(ii) B = event that the selected group consists of one boy and one girl

(iii) C = event that at least one boy is selected

Which pairs of events are mutually exclusive ?

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Solution

Let us name the boys as $B_{1}$ and $B_{2}$ , and the girls as $G_{1}, G_{2}$ and $G_{3}$ .

Then

$S = {B_{1} B_{2}, B_{1} G_{1}, B_{1} G_{2}, B_{1} G_{3}, B_{2} G_{1}, B_{2} G_{2}, B_{2} G_{3}, G_{1} G_{2}, G_{1} G_{3}, G_{2} G_{3}}$

We have

(i) $A = {G_{1} G_{2}, G_{1} G_{3}, G_{2} G_{3}}$

(ii) $B = {B_{1} G_{1}, B_{1} G_{2}, B_{1} G_{3}, B_{2} G_{1}, B_{2} G_{2}, B_{2} G_{3}}$

(iii) $C = {B_{1} G_{1}, B_{1} G_{2}, B_{1} G_{3}, B_{2} G_{1}, B_{2} G_{2}, B_{2} G_{3}, B_{1} B_{2}}$

Clearly, $A \cap B = ϕ$ and $A \cap C = ϕ$

Hence, (A, B) and (A, C) are mutually exclusive events.

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