HMMT 二月 2002 · ADV 赛 · 第 7 题

HMMT February 2002 — ADV Round — Problem 7

A manufacturer of airplane parts makes a certain engine that has a probability p of failing on any given flight. Their are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For what values of p do the two plane models have the same probability of crashing?

解析

A manufacturer of airplane parts makes a certain engine that has a probability p of failing on any given flight. Their are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For what values of p do the two plane models have the same probability of crashing? ( ) ( ) 5 5 3 2 4 5 Solution: They have the same probability of failing if p (1 − p ) + p (1 − p ) + p = 2 1 ( ) 3 2 3 2 3 2 p (1 − p ) + p , which is true iff p (6 p − 15 p + 12 p − 3) = 0. This is clearly true for p = 0. 1 We know it is true for p = 1, since both probabilities would be 1 in this case, so we know 3 2 p − 1 is a factor of 6 p − 15 p + 12 p − 3. Thus, factoring gives that the engines have the 2 2 same probability of failing if p ( p − 1)(6 p − 9 p + 3) = 0. By the quadratic formula (or by 1 1 factoring), the quadratic has roots p = , 1, so the answer is 0 , , 1 . 2 2