Question : The probabilities of solving a problem by three students A, B, and C are $\frac{3}{7}, \frac{5}{9}$ and $\frac{1}{5}$, respectively. The probability that the problem will be solved is:
Option 1: $\frac{155}{315}$
Option 2: $\frac{64}{315}$
Option 3: $\frac{251}{315}$
Option 4: $\frac{32}{315}$
Correct Answer: $\frac{251}{315}$
Solution :
Let A, B, and C respectively denote the events that the problem is solved by A, B, and C respectively and A', B', and C' respectively denote the events that the problem will not be solved.
P(A) = $\frac{3}{7}$; P(A') = $\frac{4}{7}$
P(B) = $\frac{5}{9}$; P(B') = $\frac{4}{9}$
P(C) = $\frac{1}{5}$; P(C') = $\frac{4}{5}$
The problem will be solved if even one of them solves it.
So, first we calculate the probability that is not solved = P(A') × P(B') × P (C') = $\frac{4}{7}×\frac{4}{9}×\frac{4}{5}=\frac{64}{315}$
So, The required probability = $1-\frac{64}{315}=\frac{251}{315}$
Hence, the correct answer is $\frac{251}{315}$.
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