A problem in mathematics is given to 3 students whose chances of … - Vidyadip

MCQ

A problem in mathematics is given to $3$ students whose chances of solving it are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$. What is the probability that the problem is solved?

A
$1 / 5$
B
$1 / 4$
✓
$3 / 4$
D
$2 / 3$

✓

Answer

Correct option: C.

$3 / 4$

Let $A, B, C$ be the respective events of solving the problem.
Then, $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}$ and $P(C)=\frac{1}{4}$.
Clearly $A, B, C$ are independent events and the problem is solved if atleast one student solves it.
$\therefore \text { Required probability }=P(A \cup B \cup C)$
$=1-P(\bar{A}) P(\bar{B}) P(\bar{C})$
$=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)$
$=1-\frac{1}{4}$
$=\frac{3}{4}$

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