Question

There are 3 men and 2 women. A 'Gramswachhatta Abhiyan' committee of two is to be formed:
P is the event that the committee should contain a least one woman.
Q is the event that the committee should contain one man and one woman.
R is the event that there should not be a woman in the committee.

Answer 1

There are 3 men M₁, M₂ and M₃ and 2 women W₁ and W₂.
The Gramswachhatta Abhiyan Committee of two members is to be formed.
We have:
Sample space = {M₁M₂, M₁M₃, M₂M₃, W₁W₂, M₁W₁, M₂W₁, M₃W_1, M₁W₂, M₂W₂, M₃W₂}
Or,
n(S) = 10
Now,
P is the event that the committee should have at least one woman.
Here,
P = {W₁W₂, M₁W₁, M₂W₁, M₃W₁, M₁W₂, M₂W₂, M₃W₂}
∴ n(P) = 7

Q is the event that the committee should have one man and one woman.
Here,
Q = {M₁W_1, M₂W₁, M₃W₁, M₁W₂, M₂W₂, M₃W₂}
∴ n(Q) = 6

R is the event that the committee should not have any woman.
Here,
R = {M₁M₂, M₁M₃, M₂M₃ }
∴ n(R) = 3

Now,
$\mathrm{P}\text{'}=\mathrm{S}-\mathrm{P}$ = {M₁M₂, M₁M₃, M₂M₃} = R, $\mathrm{P}\cap \mathrm{R}=\mathrm{\varphi }\mathrm{and}\mathrm{P}\cup \mathrm{R}=\mathrm{S}$
Thus, we can say that events P and R are complementary, mutually exclusive and exhaustive.
And,
$\mathrm{Q}\text{'}=\mathrm{S}-\mathrm{Q}$ = {M₁M₂, M₁M₃, M₂M₃, W₁W₂}, $\mathrm{Q}\cap \mathrm{R}=\mathrm{\varphi }$
Thus, we can say that events Q and R are mutually exclusive.