A problem in Mathematics is given to three students A, B and C. The probability of A solving the problem is 1/2 and B not solving is 1/4. The whole probability of the problem being solved is 63/64. What is the probability of solving it by C?

Concept:

Probability of compound event [(A and B) or (B and C)]:

• P [(A and B) or (B and C)] = [P (A) × P (B)] + [P (B) × P (C)]
• AND means multiply and OR means addition

​Calculation:

Given:

P (A) = 1/2, P (B') = 1/4, Let  P (C) = x

∴ Probability of A not solving the problem, P (A') = 1 - P (A) = 1 - \(\frac{1}{2}\) = \(\frac{1}{2}\)

Now, the probability that the problem is not solved at all:

= P(A') and P(B') and P(C')

= P(A') × P(B') × P(C')

= \(\frac{1}{2}~\times~\frac{1}{4}~\times~(1~-~x)\)

= \(\frac{1~-~x}{8}\)

Given that the probability that the problem is solved = \(\frac{63}{64}\)

So the probability that the problem is not solved = \(1~-~\frac{63}{64}~=~\frac{1}{64}\)

∴ \(\frac{1~-~x}{8}\) = \(\frac{1}{64}\)

⇒ x = \(1~-~\frac{1}{8}~=~~\frac{7}{8}\)

So, the probability of solving the problem by C = \(\frac{7}{8}\)