Two policemen, A and B, fire once each at the same time at an escaping convict. The probability that A hits the convict is three times the probability that B hits the convict. If the probability of the convict not getting injured is 0.5, the probability that A hits the convict is: A. 0.14 B. 0.22 C. 0.33 D. 0.80

GATE 2012 · General Aptitude · Probability · medium

Answer: P(A hits the convict) approx 0.14

1. Set up the equation using independence: Let P(B hits) = q; then P(A hits) = 3q. P(A misses) = 1-3q, P(B misses) = 1-q. Equation: (1-3q)(1-q) = 0.5
2. Expand and simplify to a quadratic: (1-3q)(1-q) = 1 - q - 3q + 3q^2 = 1 - 4q + 3q^2 = 0.5. Rearranging: 3q^2 - 4q + 0.5 = 0. Multiply by 2: 6q^2 - 8q + 1 = 0. Discriminant: 64 - 24 = 40. q = (8 +/- sqrt(40)) / 12 = (8 +/- 6.324) / 12. Two roots: q = 14.324/12 = 1.194 (rejected, > 1/3) or q = 1.676/12 = 0.1397 approx 0.14/3.
3. Find P(A hits) from the valid root: q approx 0.0466. P(A hits) = 3q approx 3 * 0.0466 = 0.1397 approx 0.14