Q66 / 180 · KCET 2021 · Mathematics

Given that, $A$ and $B$ are two events such that $P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P(A \cup B)=\frac{4}{5}$, then $P(A)$ is equal to

  • A
    $\frac{3}{10}$
  • Correct answer
    B
    $\frac{1}{2}$
  • C
    $\frac{1}{5}$
  • D
    $\frac{3}{5}$

Solution

Given, $P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P(A \cup B)=\frac{4}{5}$ $\because \quad P\left(\frac{A}{B}\right)=\frac{1}{2}$ $\Rightarrow \quad \frac{P(A \cap B)}{P(B)}=\frac{1}{2}$ $\Rightarrow \quad P(A \cap B)=\frac{1}{2} \times \frac{3}{5}=\frac{3}{10}$ Now, $\quad P(A \cup B)=\frac{4}{5}$ $\Rightarrow \quad P(A)+P(B)-P(A \cap B)=\frac{4}{5}$ $\Rightarrow \quad P(A)+\frac{3}{5}-\frac{3}{10}=\frac{4}{5}$ $\Rightarrow \quad P(A)=\frac{4}{5}+\frac{3}{10}-\frac{3}{5}$ $=\frac{8+3-6}{10}=\frac{5}{10}$ $\therefore \quad P(A)=\frac{1}{2}$

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

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