Chapter 8.6, Problem 64E

(a).

To determine

To find:

The probability that all the children are boys.

Answer to Problem 64E

The required probability is $0.0625$ .

Explanation of Solution

Given:

Assume that the probability of the birth of a child of a particular sex is $50\%$ . In a family with four children.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of boy is $\frac{1}{2}$ .

The probability of girl is $\frac{1}{2}$ .

  $\begin{array}{c}P\left(allboy\right)={\left(\frac{1}{2}\right)}^4\\ =\frac{1}{16}\\ =0.0625\end{array}$

Conclusion:

Hence, the required probability is $0.0625$ .

(b).

To determine

To find:

The probability that all the children are the same sex.

Answer to Problem 64E

The required probability is $0.125$ .

Explanation of Solution

Given:

Assume that the probability of the birth of a child of a particular sex is $50\%$ . In a family with four children.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of boy is $\frac{1}{2}$ .

The probability of girl is $\frac{1}{2}$ .

  $\begin{array}{c}P\left(allboy\right)={\left(\frac{1}{2}\right)}^4\\ =\frac{1}{16}\\ =0.0625\end{array}$

  $\begin{array}{c}P\left(allgirl\right)={\left(\frac{1}{2}\right)}^4\\ =\frac{1}{16}\\ =0.0625\end{array}$

Therefore, the probability that all four children are the same sex is:

  $\begin{array}{c}P\left(samesex\right)=2\left(0.0625\right)\\ =0.125\end{array}$

Conclusion:

Hence, the required probability is $0.125$ .

(c).

To determine

To find:

The probability that there is at least one boy.

Answer to Problem 64E

The required probability is $0.9375$ .

Explanation of Solution

Given:

Assume that the probability of the birth of a child of a particular sex is $50\%$ . In a family with four children.

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability of boy is $\frac{1}{2}$ .

The probability of girl is $\frac{1}{2}$ .

  $\begin{array}{c}P\left(allboy\right)={\left(\frac{1}{2}\right)}^4\\ =\frac{1}{16}\\ =0.0625\end{array}$

  $\begin{array}{c}P\left(allgirl\right)={\left(\frac{1}{2}\right)}^4\\ =\frac{1}{16}\\ =0.0625\end{array}$

Therefore, the probability that all four children are girls is $0.0625$ .

Thus, the probability of at least one of the four children being boy is $1-0.0625=0.9375$ .

Conclusion:

Hence, the required probability is $0.9375$ .