In these questions assume that the boys and girls are equally likely to be born.
In a two-child family, one child is a boy. What is the probability that the other child is a girl?
The probability is \(\frac{2}{3}\)
Proof
There are 4 possible two-child families:
- Boy Boy
- Boy Girl
- Girl Boy
- Girl Girl
(4) is not possible, since we know that the family has a boy. There are 3 possibilities and in 2 of the the other child is a girl.
This gives a probability of \(\frac{2}{3}\).
The unintuitive result is because the probability of a family having both a boy and a girl is twice as likely as having 2 boys (or 2 girls).
In a two-child family, the older child is a boy. What is the probability that the other child is a girl?
The probability is \(\frac{1}{2}\)
Proof
List the 4 possible two-child families. The older child is listed first, then the younger.
- Boy Boy
- Boy Girl
- Girl Boy
- Girl Girl
This time only (1) and (2) are possible since we know that the older child is a boy. In only 1 of these 2 possibilities is the other child a girl.
This gives a probability of \(\frac{1}{2}\).