Chapter 8.6, Problem 68E

(a)

To determine

To find: the probability that all the children are boys.

Answer to Problem 68E

The probability that all four children are boys is $\frac{1}{8}$ .

Explanation of Solution

Given information:

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

Calculation:

Given that the probability of the birth of a child of a particular sex is

  $\begin{array}{c}p\left(E\right)=50\%\\ =\frac{50}{100}\\ =\frac{1}{2}\\ p\left(\overline{E}\right)=1-p\left(E\right)\\ =1-\frac{1}{2}\\ =\frac{1}{2}\end{array}$

The probability that all four children are boys is

  $\begin{array}{c}p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)=\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\\ =\frac{1}{8}\end{array}$

The probability that all four children are boys is $\frac{1}{8}$ .

(a)

To determine

To find: the probability that all the children are boys.

Answer to Problem 68E

The probability that all four children are boys is $\frac{1}{8}$ .

Explanation of Solution

Given information:

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

Calculation:

Given that the probability of the birth of a child of a particular sex is

  $\begin{array}{c}p\left(E\right)=50\%\\ =\frac{50}{100}\\ =\frac{1}{2}\\ p\left(\overline{E}\right)=1-p\left(E\right)\\ =1-\frac{1}{2}\\ =\frac{1}{2}\end{array}$

The probability that all four children are boys is

  $\begin{array}{c}p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)=\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\\ =\frac{1}{8}\end{array}$

The probability that all four children are boys is $\frac{1}{8}$ .

(b)

To determine

To find: the probability that all the children are the same sex.

Answer to Problem 68E

The probability that all four children are same sex is $\frac{1}{4}$ .

Explanation of Solution

Given information:

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

Calculation:

Given that the probability of the birth of a child of a particular sex is

  $\begin{array}{c}p\left(E\right)=50\%\\ =\frac{50}{100}\\ =\frac{1}{2}\\ p\left(\overline{E}\right)=1-p\left(E\right)\\ =1-\frac{1}{2}\\ =\frac{1}{2}\end{array}$

The probability that all four children are same sex is

  $\begin{array}{c}p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)\cdot p\left(E\right)+p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)=\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}+\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\\ =\frac{1}{8}+\frac{1}{8}\\ =\frac{1}{4}\end{array}$

The probability that all four children are same sex is $\frac{1}{4}$ .

(c)

To determine

To find: the probability that there is at least one boy.

Answer to Problem 68E

The probability that there is at least one boy $\frac{7}{8}$ .

Explanation of Solution

Given information:

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

Calculation:

The probability that there is at least one boy

  $\begin{array}{l}=1-\text{The}\text{probability}\text{that}\text{all}\\ \text{=}1-p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)\cdot p\left(\overline{E}\right)\\ =1-\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\\ =1-\frac{1}{8}\\ =\frac{7}{8}\end{array}$

The probability that there is at least one boy $\frac{7}{8}$ .