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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 a girl, (ii) at least one is a girl ?

Question

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 a girl, (ii) at least one is a girl ?

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Solution

(i) The youngest is a girl:

There are four equally likely possibilities for the gender of two children: Boy-Boy, Boy-Girl, Girl-Boy, Girl-Girl. Given that the youngest is a girl, we can eliminate the Boy-Boy and Boy-Girl possibilities. This leaves us with two possibilities: Girl-Boy and Girl-Girl. The probability that both are girls (Girl-Girl) given that the youngest is a girl is therefore 1/2.

(ii) At least one is a girl:

Again, there are four equally likely possibilities for the gender of two children: Boy-Boy, Boy-Girl, Girl-Boy, Girl-Girl. Given that at least one is a girl, we can eliminate the Boy-Boy possibility. This leaves us with three possibilities: Boy-Girl, Girl-Boy, Girl-Girl. The probability that both are girls (Girl-Girl) given that at least one is a girl is therefore 1/3.

This problem has been solved

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