Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = 5/9, is :

Solution

Maths sir · 4 steps

Write exactly one as A without B plus B without A, solve the quadratic, then keep the larger root.

1. Split into two cases. Exactly one occurs means A happens and B does not, or B happens and A does not. These cases cannot happen together, so add them. Independence lets you multiply within each case.

   p(1-2p) + 2p(1-p) = 3p - 4p² = 5/9

2. Solve the quadratic. Multiply by 9 to get 36p² - 27p + 5 = 0. The discriminant is 729 - 720 = 9, so p = (27 ± 3)/72. That gives p = 5/12 or p = 1/3.
3. Check both roots. You need 2p ≤ 1, so p ≤ 1/2. Both roots pass: for p = 5/12, P(B) = 5/6.
4. So. The largest valid value is p = 5/12, which is option (d).

The idea to keep. For independent events, P(exactly one) = P(A) P(B') + P(A') P(B). When a quadratic gives two roots, check both are valid probabilities before choosing.

Answer from the NTA final answer key. Solution by PrepWiser.