Question

If $A$ and $B$ are any two events, then the probability that exactly one of them occur is

• A. $P\left(A\right)+P\left(B\right)-P(A\cap B)$
• B. $P\left(A\right)+P\left(B\right)-2P(A\cap B)$
• C. $P\left(A\right)+P\left(B\right)-P\left(AUB\right)$
• D. $P\left(A\right)+P\left(B\right)-2P\left(AUB\right)$

Answer 1

The correct option is B

$P\left(A\right)+P\left(B\right)-2P(A\cap B)$

Explanation for the correct option:

Find the required probability:

Given, $A$ and $B$ are any two events

$\therefore$The probability that exactly one of the event occurs $=P$($A$ occurs and $B$ does not occur) or $P$($A$ does not occur and $B$ occurs)

$$\begin{gathered} =P(A\cap B\text{'})+P(A\text{'}\cap B) \\ =P\left(A\right)-P(A\cap B)+P\left(B\right)–P(A\cap B) \\ =\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{+}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{-}\mathbf{2}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)} \end{gathered}$$

Hence, Option ‘B’ is Correct.