Question

From a class of $12$ girls and $18$ boys, two students are chosen randomly. What is the probability that both of them are girls?

• A. $\frac{22}{145}$
• B. $\frac{13}{15}$
• C. $\frac{1}{18}$
• D. None of these.

Answer 1

The correct option is A

$\frac{22}{145}$

Explanation for the correct option:

Calculate the probability of $2$ students selected at random are girls.

The class has 18 girls and 12 boys.

Therefore, the class has a total students of

$$\begin{gathered} =(18+12) \\ =30 \end{gathered}$$

Now, selecting $2$ girls from $30$ students in different ways

$$\begin{gathered} C_2^{30} \\ =\frac{30!}{2!} \\ =\frac{30\times 29\times 28!}{28!\times 2} \\ =435 \end{gathered}$$

Again, selecting $2$girls from a total of $12$ girls is

$$\begin{gathered} =C_2^{12} \\ =\frac{12!}{2!} \\ =\frac{12\times 11\times 10!}{10!\times 2} \\ =66 \end{gathered}$$

Therefore, the probability of $2$ students selected among a total of $30$ at random are girls

$$\begin{gathered} =\frac{66}{435} \\ =\frac{\mathbf{22}}{\mathbf{145}} \end{gathered}$$

Hence, Option (A) is the correct answer.