A problem in Mathematics is given to three students whose chances of solving it are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$, respectively. If the events of solving the problem are independent, then the probability that the problem will be solved, is

$\frac{3}{4}$

The correct answer is Option (4) → $\frac{3}{4}$ ##

Let $A, B, C$ be the respective events of solving the problem. Then,

$P(A) = \frac{1}{2}$

$P(B) = \frac{1}{3}$

and, $P(C) = \frac{1}{4}$

Here, $A, B, C$ are independent events.

Problem is solved if at least one of them solves the problem.

$\text{Required probability} = P(A \cup B \cup C)$

$= 1 - P(\bar{A})P(\bar{B})P(\bar{C})$

$= 1 - \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)$

$= 1 - \frac{1}{4}$

$= \frac{3}{4}$