HMMT 十一月 2010 · GEN2 赛 · 第 6 题

英文解析

6. [ 4 ] When flipped, a coin has a probability p of landing heads. When flipped twice, it is twice as likelyto land on the same side both times as it is to land on each side once. What is the larger possible valueof p ?
   √
   3+ 3 2 2
   Answer: The probability that the coin will land on the same side twice is p + (1 − p ) =
   2 p − 2 p + 1. The probability that the coin will land on each side once is p ( p − 1) + ( p − 1) p =26
   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 2
   √
   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