无放回抽样中的猜牌问题与生日问题
Card guessing and the birthday problem for sampling without replacement
AI总结：
本文研究允许重复数字的随机牌堆中猜牌游戏的期望正确次数，建立了最优与最差策略下的精确渐近公式，并借助Stein方法解决无放回抽样生日问题，从而回答了Diaconis等人提出的问题。
摘要：
考虑一副均匀随机的牌，牌面标有从1到n的数字，允许重复。猜测者猜测最顶部的牌，随后该牌被揭示并移除，游戏继续进行。在最优策略和最差策略下，正确猜测的期望次数是多少？我们为这两种策略建立了精确的渐近结果。对于最差情形，这回答了Diaconis、Graham、He和Spiro最近提出的一个问题，他们此前确定了正确的阶。作为证明的一部分，我们利用Stein方法研究了无放回抽样中的生日问题。
英文摘要：
Consider a uniformly random deck consisting of cards labelled by numbers from $1$ through $n$, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the game continues. What is the expected number of correct guesses under the best and worst strategies? We establish sharp asymptotics for both strategies. For the worst case, this answers a recent question of Diaconis, Graham, He and Spiro, who found the correct order. As part of the proof, we study the birthday problem for sampling without replacement using Stein's method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
