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+1 vote
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in Probability by (29.8k points)
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Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that

i. the youngest is girl

ii. at least one is a girl.

1 Answer

+2 votes
by (28.9k points)
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Best answer

Let B= boy G= girl

And let us consider, in a sample space, the first child is elder and second child is younger. 

Total possible outcome = {BB, BG, GB, GG} = 4

Let A = be the event that both the children are girls = 1

Therefore P(A) = \(\cfrac14\)

Case 1.

Let B = event that youngest is girl = {BG, GG} =2

{Since we have considered second is younger in a sample space}

Therefore P(B) = \(\cfrac24\)

And (A ∩ B) = both are girls and younger is also girl = (GG) = 1

Therefore , P (A ∩ B) = \(\cfrac14\)

We require P(\(\cfrac{A}{B}\))

\(P(\cfrac{A}{B})=\cfrac{P(A\cap B)}{P(B)}\)

\(\cfrac{\frac14}{\frac24}=\cfrac12\)

Case 2.

Let B = event that at least one is girl = {BG,GB GG} =3

{Since we have considered second is younger in a sample space}

Therefore P(B) = \(\cfrac34\)

And (A ∩ B) = both are girls and atlas one is girl = (GG) = 1

Therefore , P (A ∩ B) = \(\cfrac14\)

We require P(\(\cfrac{A}B\))

\(P(\cfrac{A}{B})=\cfrac{P(A\cap B)}{P(B)}\)

\(\cfrac{\frac14}{\frac34}\) = \(\cfrac13\)

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