Question

Let $A$ and $B$ be two events such that $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\right)P\left(B\right).$ If $0 < P\left(A\right)< 1$ and $0 < P\left(B\right)< 1$ , then $P\left(A\cup B\right)^{'}=$

• A. 1-P(A)
• B. 1-P(A')
• C. 1-P(A)P(B)
• D. [1-P(A)]P(B')

Answer 1

Answer: (D)

[1-P(A)]P(B')

Answer 2

Given, $P(A \cup B)=P(A)+P(B)-P($ A) $P(B)$
$\therefore P(A \cup B)^{'}=1-P(A \cup B)$
$=1-[P(A)+P(B)-P(A) P(B)]$
$=1[1-P(A)]-P(B)[1-P(A)]$
$=[1-P(A)][1-P(B)]$
$=[1-P(A)] P\left(B^{'}\right)$