Question

Suppose that a family has $5$ children. Also, suppose that the probability of having a girl is $\frac{1}{2}$. Find the probabilities that the family has the following children.
Exactly $3$ girls and $2$ boys

Answer 1

Finding the probability of exactly $3$ girls and $2$ boys:

Given,

The family has total children $n=5$
Probability of success of having girl $p=\frac{1}{2}=0.5$
Probability of failure of having girl or success of having boys $q=1-0.5=0.5$

Probability of having exactly $3$ girls and $2$ boys ${=}^5{C}_3{\left(p\right)}^3{\left(q\right)}^{5-3}$
$$\begin{gathered} =10\times {(0.5)}^{3+2} \\ =10\times \left(0.03125\right) \\ =0.3125 \end{gathered}$$

Hence, the required probability is $0.3125$.