# Probability: A Family of Five Children

**Problem 1**

A family with 5 children is selected at random, what is the probability that all are girls or all are boys?

**Solution 1**

$P_G = \left( \dfrac{1}{2} \right)^5 = \dfrac{1}{32}$

All are boys:

$P_B = \left( \dfrac{1}{2} \right)^5 = \dfrac{1}{32}$

All are girls or all are boys:

$P = P_G + P_B = \dfrac{1}{32} + \dfrac{1}{32}$

$P = \dfrac{1}{16}$ ← *answer*

**Problem 2**

A family with 5 children is selected at random, what is the probability that there are 3 boys and 2 girls?

**Solution 2**

$P = \dfrac{5}{16}$ ← *answer*

**Note:**

= among the 5 children, 2 are girls

= in 5 children, 3 are boys and 2 are girls.

In equation, we can write it receptively as...

${^5}C_3 = {^5}C_2 = \dfrac{5!}{3! \, 2!}$

**Problem 3**

A family with 5 children is selected at random, what is the probability that at least one child is a boy?

**Solution 3**

$P = \dfrac{31}{32}$ ← *answer*

**Preferred Solution**

$P = 1 - Q_{\text{all are girls}} = 1 - \left( \dfrac{1}{2} \right)^5$

$P = \dfrac{31}{32}$ ← *answer*