Question

A die is thrown :
$P$ is the event of getting an odd number.
$Q$ is the event of getting an even number.
$R$ is the event of getting a prime number.

Which of the following pairs is mutually exclusive?

• A. $P,Q$
• B. $Q,R$
• C. $P,R$
• D. None of these

Answer 1

Answer: (A)

$P,Q$

Let $S$ be the sample space

$S=\{1,2,3,4,5,6\}$

$n\left(S\right)=6$

$P$ is the event of getting an odd number

$\therefore P=\{1,3,5\}⟹n\left(P\right)=3$

$Q$ is the event of getting an even number

$\therefore Q=\{2,4,6\}⟹n\left(Q\right)=3$

$R$ is the vent of getting a prime number

$\therefore R=\{2,3,5\}⟹n\left(R\right)=3$

The set $S$ contains two types of numbers even and odd

We have, $P=S-Q$ and $Q=S-P$
$\therefore n\left(P\right)$ and $n\left(Q\right)$ both are complementary.
Two events are mutually exclusive if they are disjoint

Now, $P\cap Q=\varphi$

$⟹P$ and $Q$ are disjoint i.e. they don't have common elements.

Hence, $P$ and $Q$ are mutually exclusive.