HMMT 二月 2002 · ADV 赛 · 第 7 题

英文解析

7. A manufacturer of airplane parts makes a certain engine that has a probability p offailing 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
   ( )
   2 3 2 3 23
   p (1 − p ) + p , which is true iff p (6 p − 15 p + 12 p − 3) = 0. This is clearly true for p = 0.
   We know it is true for p = 1, since both probabilities would be 1 in this case, so we know 1
   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