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
Answer & explanation
Correct answer: option 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.
$\text{Required probability} = P(A \cup B \cup C)$
$= 1 - P(\bar{A})P(\bar{B})P(\bar{C})$
$= 1 - \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)$
$= 1 - \frac{1}{4}$
$= \frac{3}{4}$