Q224 / 270 · NDA 2020 · Mathematics

If $A$ and $B$ are two events such that $P(A)=0.6, P(B)=0.5$ and $P(A \cap B)=0.4$, then consider the following statements : 1. $\quad P(\bar{A} \cup B)=0.9$. 2. $P(\bar{B} \mid \bar{A})=0.6$. Which of the above statements is/are correct?

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

Solution

$\begin{aligned} & \text { } P(\bar{A})=1-P(A)=1-0.6=0.4 \\ & P(\bar{B})=1-P(B)=1-0.5=0.5 \end{aligned}$ Now, $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ $=0.6+0.5-0.4=0.7$ (1) Now $P(\bar{A} \cup B)=P(\bar{A})+P(B)-P(\bar{A} \cap B)$ $\begin{aligned} & =P(\bar{A})+P(B)-\{P(B)-P(A \cap B)\} \\ = & 0.4+0.5-\{0.5-0.4\}=0.8 \end{aligned}$ Hence, statement (1) is not correct. $\begin{aligned} & \text { (2) } P(\bar{B} \mid \bar{A})=\frac{P(\bar{B} \cap \bar{A})}{P(\bar{A})}=\frac{P(B \cup A)^{\prime}}{P(\bar{A})}=\frac{1-P(A \cup B)}{P(\bar{A})} \\ & =\frac{1-0.7}{0.4}=0.75 . \end{aligned}$ Hence, statement (2) is not correct.

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

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