Problem 3

Explanation

Let $B$ be a boy and $G$ be a girl.

The sample space for a family with $3$ children (without conditions) has $2^3 = 8$ possibilities:

It is given that "at least one child is a girl". This means the event "all children are boys" ($BBB$) is impossible.

The new sample space ($S$) consists of all possibilities except $BBB$:

The number of elements in the sample space is $n(S) = 7$.

The expected event ($A$) is having $2$ girls and $1$ boy.

From the sample space $S$, the outcomes that satisfy this condition are:

The number of expected outcomes is $n(A) = 3$.

The probability of the event is:

Therefore, the probability that the family has $2$ girls and $1$ boy is $\frac{3}{7}$.