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Question

Consider the following events for a family with children
$$A=\left \{of\ both\ sexes  \right \}$$; $$B=\left \{at\ most\ one\ boy  \right \}$$
In which of the following $$(are/is)$$ the events $$A$$ and $$B$$ are independent
(a)If a family has $$3$$ children (b) If a family has $$2$$ children
Assume that the birth of a boy or a girl is equally likely mutually exclusive and exhaustive


Solution

(A)
$$P(A)=\dfrac{n(S)n(not\ both\ sexes)}{n(S)}=\dfrac{2^{3}-2}{2^{3}}=\dfrac{3}{4}$$
favourable outcomes for not both sexes=(GGG), (BBB)$$
$$P(B)=\dfrac{4}{8}=\dfrac{1}{2}\left[(GGG), (BGG), (GBG), (GGB)\right]$$
$$P(A\cap B)=\dfrac{3}{8}\left[(GBG), (BGG), (GGB)\right]$$
$$P(A). P(B)=\dfrac{3}{4}-\dfrac{1}{2}=\dfrac{3}{8}=P(A\cap B)\Rightarrow A$$ & $$B$$ are independent
(B)
$$P(A)=\dfrac{4-2}{4}=\dfrac{1}{2}$$ [Non-favourable outcomes $$(GG), (BB)$$]
$$P(B)=\dfrac{3}{4}[(GG), (GB), (BG)]$$
$$P(A\cap B)=\dfrac{2}{4}=\dfrac{1}{2} [(GB, (BG)]$$
$$P(A). P(B)=\dfrac{1}{2}\times \dfrac{3}{4}=\dfrac{3}{8}\neq P(A\cap B)\Rightarrow A$$ & $$B$$ are not independent










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