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 Exercise 33 lists the sample space for a couple having three children. After identifying the sample space for a couple having four children, find the probability of getting three girls and one boy (in any order).

Short answer

The probability of having three girls and one boy out of four children is equal to 0.25.

Step-by-step solution

Given information

A couple has four children. The genders of the four children need to be listed.

Identify the sample space 

Sample spaceis the complete enumeration of all outcomes of an event.

On any particular order of childbirth, there are two possible outcomes, g(girl) or b(boy).

Therefore, on fourbirths, there would be ${\left(2\right)}^4=16$possible combinations.

Let S be the sample space for the gender of four children as shown below:

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

Compute the probability of an event with three girls and one boy

Theprobability of an event is a number that gives a sense of the likelihood of an event.

It has the following formula:

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

The total number of gender combinations = 16

The number of combinations that have three girls and one boy is equal to four. They are (gggb, ggbg, gbgg, bggg).

The probability of three girls and one boy (in any order) out of four children is calculated as shown:

$$\begin{gathered} P\left(\text{3}\text{girls and}\text{1}\text{boy}\right)=\frac{4}{16} \\ =0.25 \end{gathered}$$

Therefore, the probability of three girls and one boy (in any order) out of four children is equal to 0.25.