Question

$A,B$ and $C$ are any three events. If $P\left(S\right)$ denotes the probability of $S$ happening, then $P(A\cap (BUC\left)\right)$ is

• A. $P\left(A\right)+P\left(B\right)+P\left(C\right)-P(A\cap B)-P(A\cap C)$
• B. $P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(B\right)P\left(C\right)$
• C. $P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$
• D. None of these

Answer 1

The correct option is C

$P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$

Explanation for the correct option:

Find the value of $\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{(}\mathbf{BUC}\mathbf{\left)}\mathbf{\right)}$

$P(A\cap (BUC\left)\right)$ $=$$P\left[\right(A\cap B\left)U\right(A\cap C\left)\right]$

Use the formula $\mathit{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}$

$=$$P(A\cap B)+P(A\cap C)–P\left(\right[(A\cap B)\cap (A\cap C))$

$=$$P(A\cap B)+P(A\cap C)–P(A\cap B\cap C)$

Hence, option ‘C’ is correct.