Question

At Least One. In Exercises 5–12, find the probability.

Three Girls Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)

Short answer

The probability of having at least one girl out of three children is equal to 0.875.

Step-by-step solution

Given information

A couple has three children.

The chances of having a girl or a boy are even.

Define the concept of probability for the event of “At least One”

In the event of selecting elements from a sample space,the probability of getting “at least one” of a specific type of item is the probability of gettingone or more than one of that item.

The probability of selecting at least one of a unit is the complementary event of selecting none of those.

It also follows the property:

$P\left(\text{at}\text{least}\text{one}\right)=1-P\left(\text{none}\right)$

Identify the sample space of three children

The sample space for the gender of the three children is as follows:

S = {bbb,bbg,bgb,gbb,bgg,gbg,ggb,ggg}

Here, b represents the male gender, whereas g represents the female gender.

The probability of having at least one girl denotes the probability of having one or more than one girl.

It can also be written as one minus the probability of having no girl.

The number of favorable outcomes of having no girl = 1.

The total number of outcomes = 8.

$$\begin{gathered} P\left(\text{at}\text{least}\text{1}\text{girl}\right)=1-P\left(\text{no}\text{girl}\right) \\ =1-\frac{1}{8} \\ =\frac{7}{8} \\ =0.875 \end{gathered}$$

Therefore, the probability of having at least one girl is equal to 0.875.