The probability that $A$ hits the target is $\frac{1}{3}$ and the probability that $B$ hits it, is $\frac{2}{5}$. If both try to hit the target independently, find the probability that the target is hit.

$\frac{3}{5}$

The correct answer is Option (2) → $\frac{3}{5}$ ##

$P(A) = P(A \text{ hits target}) = \frac{1}{3}$

$P(B) = P(B \text{ hits target}) = \frac{2}{5}$

$\text{Now, }P(A \cup B) = P(\text{target will be hit})$

$= P(A) + P(B) - P(A \cap B)$

$= P(A) + P(B) - P(A) \cdot P(B) \quad [∵A \text{ and } B \text{ are independent}]$

$= \frac{1}{3} + \frac{2}{5} - \left( \frac{1}{3} \cdot \frac{2}{5} \right) = \frac{5 + 6 - 2}{15}$

$= \frac{9}{15} = \frac{3}{5} $