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Question
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is ______.
Options
`1/2`
`1/3`
`2/3`
`4/7`
Solution
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is `4/7`.
Explanation:
Let G denotes the girl and B denotes the boy of the given family.
So, n(S) = {(BGG), (GBG), (GGB), (GBB), (BGB), (BBG), (BBB), (GGG)}
Let E1 be the event that the family has alteast one girl.
∴ E1 = {(BGG), (GBG), (GGB), (GBB), (BGB), (BBG), (GGG)}
Let E2 be the event that the eldest child is a girl.
∴ E2 = {(GBG), (GGB), (GBB), (GGG)}
(E1 ∩ E2) = {(GBB), (GGB), (GBG), (GGG)}
∴ `"P"("E"_2/"E"_1) = ("P"("E"_1 ∩ "E"_2))/("P"("E"_1))`
= `(4/8)/(7/8)`
= `4/7`
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