Question

given a family with two children, find the prabability that both are boys, given that at least one is a boy.

Answer 1

We shall denote by A and B the following events:

Event A: both children are boys;

Event B: at least one child is a boy;

The aim is to compute the conditional probability $p(A|B)$. It is given by (see https://en.wikipedia.org/wiki/Conditional_probability) :

$p(A|B)= \frac{p(A\cap B)}{p(B)}.$

We point out that event $A\cap B$ is: both children are boys;

We assume that the probability that one child is a boy is 0.5 and treat children in the same way as independent events. By direct computations we have:

$p(A\cap B)=0.5^2=0.25$

$p(B)=1-0.25=0.75$

$p(A|B)=\frac{0.25}{0.75}=\frac13\approx0.3333$ (it is rounded to four decimal places)

Answer: 0.3333 (it is rounded to 4 decimal places)