Q51 / 170 · Manipal (MET) 2020 · Mathematics

If $A$ and $B$ are two such events that $P(A \cup B)^{\prime}=P(A \cap B)$, then which of the following is true?

  • A
    $P(A)+P(B)=0$
  • B
    $P(A)+P(B)=P(A) P(B / A)$
  • Correct answer
    C
    $P(A)+P(B)=2 P(A) P(B / A)$
  • D
    None of the above

Solution

Given, $P(A \cup B)^{\prime}=P(A \cap B)$ $\Rightarrow P(A)+P(B)-P(A \cap B)=P(A \cap B)$ $\Rightarrow \quad P(A)+P(B)=2 P(A \cap B)$ $\Rightarrow \quad P(A)+P(B)=2 P(A \cap B)$ $\Rightarrow \quad P(A)+P(B)=2 \times P(A) P(B / A)$ $\left[\because P(B / A)=\frac{P(A \cap B)}{P(A)}\right]$

Hence, the correct answer is option: 'C'.

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