HMMT 二月 2002 · ADV 赛 · 第 4 题

英文解析

4. Two fair coins are simultaneously flipped. This is done repeatedly until at least oneof the coins comes up heads, at which point the process stops. What is the probability that the other coin also came up heads on this last flip?
   Solution: 1 / 3 . Let the desired probability be p . There is a 1 / 4 chance that both coinswill come up heads on the first toss. Otherwise, both can come up heads simultaneouslyonly if both are tails on the first toss, and then the process restarts as if from the beginning;
   thus this situation occurs with probability p/ 4. Thus p = 1 / 4 + p/ 4; solving, p = 1 / 3.
   Alternate Solution: The desired event is equivalent to both coins coming up tails for nsuccessive turns (for some n ≥ 0), then both coins coming up heads. For any fixed value ofn +11
   n , the probability of this occurring is 1 / 4 . Since all these events are disjoint, the total
   2 3
   probability is 1 / 4 + 1 / 4 + 1 / 4 + · · · = 1 / 3.