Chapter 11.4, Problem 20PPS

To determine

What is the probability that they will have $2$ boys?

Answer to Problem 20PPS

  $\boxed{0.375or37.5\%}$

Explanation of Solution

Given information:

Mr. and Mrs. Davis are planning to have $3$ children. The probability of each child a being a boy is $50\%$ . What is the probability that they will have $2$ boys?

Calculation:

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure which is equal to,

  $q=1-p$

Hence the probability of each chid being a boy is $50\%$ , so,

  $\begin{array}{l}p=50\%\\ =0.5\end{array}$

The probability of failure is,

  $q=1-p$

  $=1-0.5$

  $=0.5$

Substitute $p=0.5,q=0.5,k=2,n=3$ in the above formula to find the probability will have $2$ boys is,

  $\begin{array}{l}P(2){=}^3{C}_2{(}^{0.5}{(}^{0.5}\\ =3{(}^{0.5}{(}^{0.5}\\ =0.375\\ =37.5\%\end{array}$

Hence, the probability that the couple will have $2$ boys is,

  $\boxed{0.375or37.5\%}$