Question

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.)

Answer 1

Finding the required probability:

A couple can only give birth to either a girl or a boy.

Now, we are told that boys and girls are equally likely.

This can be expressed as,

Probability of getting a boy is equal to $\frac{1}{2}$

Probability of getting a girl is equal to $\frac{1}{2}$

If they have three children, we can say that the probability of not having any girl will be expressed as:

$$\begin{gathered} P\left(\text{No.of girls}\right)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} \\ =\frac{1}{8} \end{gathered}$$

Then, the probability of getting at least one to be a girl can be expressed as:

$$\begin{gathered} \mathrm{P}\left(\text{at least 1 to be a girl}\right)=1-\mathrm{P}\left(\text{not having any girl}\right) \\ =1-\frac{1}{8} \\ =\frac{7}{8} \end{gathered}$$

Hence, the probability of getting at least one to be a girl is $\frac{7}{8}$ or $0.875$.