Question

Probability from a Sample Space. In Exercises 33–36, use the given sample space or construct the required sample space to find the indicated probability.

Four Children Using the same sample space and assumption from Exercise 35, find the probability that when a couple has four children, all four are of the same gender.

Short answer

The probability of having four children of the same gender is equal to 0.125.

Step-by-step solution

Given information

A couple has four children.

Define sample space and probability 

Theprobability of an eventis a number that quantifies the possible occurrence of an event.

Its formula is written below:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{that}\text{favour}\text{A}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

Sample space is exhaustive of all possible events.

Compute the probability that all children are of the same gender

Refer to Exercise 35; the sample space has 16 outcomes shown below.

The total number of gender combinations = 16.

The sample space of the genders of four children to be given as follows:

S=(bbbb,bbbg,bbgb,bgbb,gbbb,bbgg,bgbg,gbgb,ggbb,bggb,gbbg,gggb,ggbg,gbgg,bggg,gggg)

The number of combinations that have the same gender for all four children is two. They are (bbbb, gggg).

The probability of event E, same-gender for all four children, is calculated as shown below:

$$\begin{gathered} P\left(E\right)=\frac{2}{16} \\ =0.125 \end{gathered}$$

Therefore, the probability of having the same gender for all four children is equal to 0.125.