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 :
a) At least one girl.
b) One boy and one girl
c) Only boys

Answer 1

Let,
the three boys be b₁, b₂, b₃ and
the two girls be g₁ and g₂.

A committee of two is to be formed.
The Sample space(S) is
S = {(b₁, b₂), (b₁, b₃), (b₁,g₁), (b_1, g₂), (b₂, b₃), (b_2, g₁), (b₂, g₂), 
        (b_{3 ,} g₁), (b₃, g₂), (g_1, g₂)}
⇒ n(S)=10

a) Let B be event that the committee contains only one girls.
B = { (b₁,g₁), (b_1, g₂), (b_2, g₁), (b₂, g₂), (b_{3 ,} g₁), (b₃, g₂), (g_1, g₂)}
⇒ n(B) = 7

`P(B)=(n(B))/(n(S))=7/10`

b) Let C be the event that the committee contains one boy and one girl.
C = { (b₁,g₁), (b_1, g₂), (b_2, g₁), (b₂, g₂), (b_{3 ,} g₁), (b₃, g₂)}
⇒ n(C)=6

`P(C)=(n(C))/(n(S))=6/10 = 3/5`

c) Let D be the event that the committee contains only boys.
D = { (b₁, b₂), (b₁, b₃), (b₂, b₃) }
⇒ n(D) = 3

`P(D)=(n(D))/(n(S))=3/10`