Explanation
Concept:
- P(not E) = 1 - P(E).
- Probability of a Compound Event [(A and B) or (B and C)] is calculated as:
P[(A and B) or (B and C)] = [P(A) × P(B)] + [P(C) × P(D)]
('and' means 'x' and 'or' means '+')
Calculations:
It is given that P(A) = 1/2 and P(not B) = 1/4. Let's say that the probability of C solving the problem is P(C) = x.
∴ Probability of A not solving the problem = P(not A) = 1 - P(A) = 1 - 1/2 = 1/2.
And, probability of C not solving the problem = P(not C) = 1 - P(C) = 1 - x.
Now, the probability that the problem is not solved at all:
= P(not A) AND P(not B) AND P(not C)
= P(not A) × P(not B) × P(not C)
= 1/2 × 1/4 × (1 - x)
= 1 - x/8.
And, the probability that the problem is solved = 1 - Probability that the problem is not solved at all = 63/64
⇒ 1 - 1 - x/8 = 63/64
⇒ 1 - x/8 = 1 - 63/64 = 1/64
⇒ 1 - x = 1/8
⇒ x = 1 - 1/8 = 7/8.
∴ The probability of C solving the problem is P(C) = x = 7/8.
