Q270 / 270 · NDA 2021 (Phase 1) · Mathematics

If $A$ and $B$ are two events such that $P(A)=\frac{3}{4}$ and $P(B)=$ $\frac{5}{8}$, then consider the following statements : 1. The minimum value of $P(A \cup B)$ is $\frac{3}{4}$. 2. The maximum value of $P(A \cap B)$ is $\frac{5}{8}$. Which of the above statements is/are correct?

  • A
    $1$ only
  • B
    $2$ only
  • Correct answer
    C
    Both $1$ and $2$
  • D
    Neither $1$ nor $2$

Solution

$\begin{aligned} & P(A)=\frac{3}{4} \text { and } P(B)=\frac{5}{8} \\ & P(A \cup B)=P(A)+P(B)-P(A \cap B)\end{aligned}$ $P(A \cup B)$ is $\min$ when $P(A \cap B)$ is $\max$ i.e. $\frac{5}{8}$ $P(A \cup B)_{\min }=\frac{3}{4}+\frac{5}{8}-\frac{5}{8}=\frac{3}{4}$ Hence, both (1) and (2) are correct.

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

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