A problem in Mathematics is given to three students whose chances… - Vidyadip

MCQ

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 their solving the problem are independent then the probability that the problem will be solved, is

A
$\frac{1}{4}$
B
$\frac{1}{3}$
C
$\frac{1}{2}$
D
$\frac{3}{4}$

✓

Answer

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.
Required probability is $=P(A \cup B \cup C)=1-P(\bar{A}) P(\bar{B}) P(\bar{C})$
$
=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)=1-\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4}=1-\frac{1}{4}=\frac{3}{4} \text {. }$

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