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 ______.

विकल्प

• ${\displaystyle \frac{1}{2}}$

• ${\displaystyle \frac{1}{3}}$

• ${\displaystyle \frac{2}{3}}$

• ${\displaystyle \frac{4}{7}}$

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 ${\displaystyle \frac{4}{7}}$.

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)}

∴ ${\displaystyle \text{P}\left(\frac{{\text{E}}_2}{{\text{E}}_1}\right)=\frac{\text{P}({\text{E}}_1\cap {\text{E}}_2)}{\text{P}\left({\text{E}}_1\right)}}$

= ${\displaystyle \frac{\frac{4}{8}}{\frac{7}{8}}}$

= ${\displaystyle \frac{4}{7}}$