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`

Answer 1

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`