HMMT 二月 2002 · GEN2 赛 · 第 8 题
HMMT February 2002 — GEN2 Round — Problem 8
题目详情
- Two fair coins are simultaneously flipped. This is done repeatedly until at least one of 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?
解析
- Two fair coins are simultaneously flipped. This is done repeatedly until at least one of 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 coins will come up heads on the first toss. Otherwise, both can come up heads simultaneously only 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 n successive turns (for some n ≥ 0), then both coins coming up heads. For any fixed value of n +1 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.