Question

A family has $5$ children. The probability of having a girl is $\frac{1}{2}$. What is the probability of having exactly $3$ girls and $2$ boys?

Answer 1

Finding the required probability:

Let $X$ denote the number of girls in $n=5$, where each has a probability $p=0.5$ ,

We can solve it by using the binomial coefficients,

$X~\left(Binomial\right)(n=5,p=0.5)$

The probability of $x$ number of girls is given by the formula:

$P(X=x)={}_{}{}^{n}C_x{p}^x{(1-p)}^{n-x}$

Substituting $P(X=3),n=5,p=0.5$

$$\begin{gathered} P(X=3)={}_{}{}^{5}C_{3}0.5^3{(1-0.5)}^{5-3} \\ =10\times 0.5^3\times 0.5^2 \\ =0.313 \end{gathered}$$

Therefore, the probability of having exactly $3$ girls and $2$ boys is $0.313$.