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.

Three Children Using the same sample space and assumption from Exercise 33, find the probability that when a couple has three children, there are exactly two girls.

Short answer

The probability of having exactly two girls when a couple has three children is equal to 0.375.

Step-by-step solution

Given information

The combinations of the gender of children when a couple has three children is given as (bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg), where b means a boy, and g means a girl.

Probability and sample space

Theprobability of an event is a number that gives a sense of the plausibility of an event. It has the following formula:

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

Compute the probability for exactly two girls 

Let S be the sample space for the genders of three children as shown below:

S =(bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg), where b denotes a boy, and g denotes a girl.

The total number of gender combinations = 8.

The number of combinations that have exactly two girls is equal to three. They are (bgg,gbg,ggb).

The probability of event F, that is, two girls among three children is calculated as shown:

$$\begin{gathered} P\left(F\right)=\frac{3}{8} \\ =0.375 \end{gathered}$$

Therefore, the probability of exactly two girls out of three children is equal to 0.375.