Question

There are three boys and two girls. A committee of two is to be formed, Find the probability of events that the committee contains:
(i) at least one girl.
(ii) one boy and one girl.
(iii) only boys.

Answer 1

There are 3 boys (B₁, B₂ and B₃) and 2 girls (G₁ and G₂). A committee of two members is to be formed.
$\therefore$ Sample space = {B₁B₂, B₁B₃, B₂B₃, G₁G₂, B₁G₁, B₂G₁, B₃G₁, B₁G₂, B₂G₂, B₃G₂}
Or,
n(S) = 10

(i) Let A be the event that the committee has at least one girl.
A = {G₁G₂, B₁G₁, B₂G₁, B₃G₁, B₁G₂, B₂G₂, B₃G₂}
∴ n(A) = 7
Thus, we have:
$$\begin{gathered} \mathrm{P}\left(\mathrm{A}\right)=\frac{n\left(\mathrm{A}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{7}{10} \end{gathered}$$

(ii) Let B be the event that the committee has one boy and one girl.
B = {B₁G₁, B₂G₁, B₃G₁, B₁G₂, B₂G₂, B₃G₂}
∴ n(B) = 6
Thus, we have:
$$\begin{gathered} \mathrm{P}\left(\mathrm{B}\right)=\frac{n\left(\mathrm{B}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{B}\right)=\frac{6}{10}=\frac{3}{5} \end{gathered}$$

(iii) Let C be the event that the committee has only boys.
C = {B₁B₂, B₁B₃, B₂B₃}
∴ n(C) = 3
Thus, we have:
$$\begin{gathered} \mathrm{P}\left(\mathrm{C}\right)=\frac{n\left(\mathrm{C}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{C}\right)=\frac{3}{10} \end{gathered}$$