Q108 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

If $\mathrm{A}$ and $\mathrm{B}$ are two events in a random experiment such that $\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})$ then

  • A
    $P(A)+P(B)=1$
  • Correct answer
    B
    $P(A)=P(B)$
  • C
    $P(A)+P(B)>1$
  • D
    $P(A)=0, P(B)=1$

Solution

$P(A)+P(B)=2 P(A \cap B)$ $\therefore \quad P(A)+P(B)-P(A \cap B)=P(A \cap B)$ But we know that $\begin{aligned} & P(A)+P(B)-P(A \cap B)=P(A \cup B) \\ & \therefore \quad P(A \cup B)=P(A \cap B) \\ & \Rightarrow \quad P(A)=P(B) . \end{aligned}$

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

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