HMMT 十一月 2010 · GEN2 赛 · 第 9 题

英文解析

9. [ 6 ] Newton and Leibniz are playing a game with a coin that comes up heads with probability p . Theytake turns flipping the coin until one of them wins with Newton going first. Newton wins if he flips aheads and Leibniz wins if he flips a tails. Given that Newton and Leibniz each win the game half ofthe time, what is the probability p ?
   √
   3 − 5
   Answer: The probability that Newton will win on the first flip is p . The probability that
   Newton will win on the third flip is (1 − p ) p , since the first flip must be tails, the second must be heads,22
   and the third flip must be heads. By the same logic, the probability Newton will win on the (2 n + 1)stn n +1 2 2 3
   flip is (1 − p ) ( p ) . Thus, we have an infinite geometric sequence p + (1 − p ) p + (1 − p ) p + . . .
   √
   1 3 − 5 pwhich equals . We are given that this sum must equal , so 1 − p + p = 2 p , so p = (the 2
   1 − p (1 − p ) 2 2
   other solution is greater than 1).
   Theme Round
   2 1