Q77 / 160 · TS EAMCET 2020 (10 Sep Shift 1) · Mathematics

If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

  • Correct answer
    A
    $\frac{5}{12}$
  • B
    $\frac{3}{8}$
  • C
    $\frac{4}{5}$
  • D
    $\frac{5}{4}$

Solution

We have, $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$ $\begin{aligned} & \text { We know that, } P(A \cup B)=P(A)+P(B)-P(A \cap B) \\ & \Rightarrow P(B)-P(A \cap B)=P(A \cup B)-P(A) \\ & \Rightarrow P(\bar{A} \cap B)=P(A \cup B)-P(A) \\ & =\frac{3}{4}-\left(1-\frac{2}{3}\right)=\frac{3}{4}-\frac{1}{3}=\frac{5}{12} \\ & \end{aligned}$

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