Question

A problem in statistics is given to three students $P$, $Q$ and $R$. Their chances of solving the problem are $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$ respectively. If all of them try independently, then the probability that the problem is solved is

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

Hint:

When multiple independent events are involved, it is often easier to calculate the probability of failure first and subtract it from 1.

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
Each student attempts the problem independently. The probability that the problem is solved means at least one student solves it.
Step 2: Finding the probability that none of them solves the problem.
\[ P(\text{P fails}) = 1 - \frac{1}{2} = \frac{1}{2} \] \[ P(\text{Q fails}) = 1 - \frac{1}{3} = \frac{2}{3} \] \[ P(\text{R fails}) = 1 - \frac{1}{4} = \frac{3}{4} \] \[ P(\text{none solves}) = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{4} \]
Step 3: Finding the probability that the problem is solved.
\[ P(\text{problem solved}) = 1 - \frac{1}{4} = \frac{3}{4} \]
Step 4: Conclusion.
The probability that the problem is solved is $\frac{3}{4}$.