HMMT 十一月 2010 · GEN2 赛 · 第 6 题
HMMT November 2010 — GEN2 Round — Problem 6
题目详情
- [ 4 ] When flipped, a coin has a probability p of landing heads. When flipped twice, it is twice as likely to land on the same side both times as it is to land on each side once. What is the larger possible value of p ?
解析
- [ 4 ] When flipped, a coin has a probability p of landing heads. When flipped twice, it is twice as likely to land on the same side both times as it is to land on each side once. What is the larger possible value of p ? √ 3+ 3 2 2 Answer: The probability that the coin will land on the same side twice is p + (1 − p ) = 6 2 2 p − 2 p + 1. The probability that the coin will land on each side once is p ( p − 1) + ( p − 1) p = 2 2 p (1 − p ) = 2 p − 2 p . We are told that it is twice as likely to land on the same side both times, so √ 3 ± 3 2 2 2 2 2 p − 2 p + 1 = 2(2 p − 2 p ) = 4 p − 4 p . Solving, we get 6 p − 6 p + 1 = 0, so p = . The larger 6 √ 3+ 3 solution is p = . 6