硬币配对 II
Coin Pair II
题目详情
你同时抛 4 枚公平硬币并观察结果。之后你可以无限次地选择任意两枚硬币“再抛一次”（每次必须选两枚，不能只选一枚）。
若你最优操作，最终出现的正面数期望是多少？
Four fair coins appear in front of you. You flip all four at once and observe the outcomes of the coins. After seeing the outcomes, you may flip any pair of coins again unlimited amounts of times. You may not flip a single coin without flipping another. Assuming optimal play, find the expected number of heads that appear.
解析
把“全正面 HHHH”视为吸收态。由于你可以反复选择两枚重掷，而每次选择都有正概率让正面数变多，因此该马尔可夫过程以概率 1 最终会到达全正面。
所以最终正面数期望为 4。
Original Explanation
For each pair we flip again, we have a positive probability of obtaining 2 heads. If we are to flip every pair of coins, regardless of the initial outcome, enough times, we would end with all heads. Therefore, the answer is just 4, as with probability 1, we will obtain at some point.