1975 USAMO 第 5 题

A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$.

一副 $n$ 扑克牌包含三张 A，随机洗牌(假设所有可能的牌分布的可能性相同)。然后牌从顶部一张一张地翻开，直到第二张 A 出现。证明要翻开的卡片的预期(平均)数量为 $(n+1)/2$。