Q236 / 270 · NDA 2013 (Phase 2) · Mathematics

If $\mathrm{P}(\mathrm{A})=\frac{2}{3}, \mathrm{P}(\mathrm{B})=\frac{2}{5}$, and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5}$, then what is $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ equal to?

  • A
    $\frac{3}{5}$
  • B
    $\frac{5}{11}$
  • Correct answer
    C
    $\frac{1}{3}$
  • D
    None of the above

Solution

$\begin{aligned} & \mathrm{P}(\mathrm{A} \cup \mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\ & =\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5} \\ & =\frac{2}{3}+\frac{2}{5}-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5} \\ & \therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{3}\end{aligned}$

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

Relevant Questions