The last success problem is an optimal stopping problem that aims to maximize the probability of stopping on the last success in a sequence of $n$ Bernoulli trials. In a typical setting where complete information about the distributions is available, Bruss provided an optimal stopping policy ensuring a winning probability of $1/e$. However, assuming complete knowledge of the distributions is unrealistic in many practical applications. In this paper, we investigate a variant of the last success problem where we have single-sample access from each distribution instead of having comprehensive knowledge of the distributions. Nuti and Vondr\'{a}k demonstrated that a winning probability exceeding $1/4$ is unachievable for this setting, but it remains unknown whether a stopping policy that meets this bound exists. We reveal that Bruss's policy, when applied with the estimated success probabilities, cannot ensure a winning probability greater than $(1-e^{-4})/4\approx 0.2454~(< 1/4)$, irrespective of the estimations from the given samples. Nevertheless, we demonstrate that by setting the threshold the second-to-last success in samples and stopping on the first success observed \emph{after} this threshold, a winning probability of $1/4$ can be guaranteed.

翻译：最后成功问题是一个最优停止问题，旨在最大化在$n$次伯努利试验序列中停止在最后一次成功上的概率。在完全掌握分布信息的典型设定下，Bruss提供了一种最优停止策略，确保获胜概率为$1/e$。然而，在许多实际应用中，假设完全了解分布是不现实的。本文研究了最后成功问题的一个变体：我们仅能通过单次样本访问每个分布，而非全面掌握分布信息。Nuti和Vondrák证明，在该设定下获胜概率无法超过$1/4$，但尚不清楚是否存在达到这一界限的停止策略。我们揭示：Bruss策略在使用估计成功概率时，无论基于给定样本的估计值如何，其获胜概率均无法超过$(1-e^{-4})/4\approx 0.2454~(< 1/4)$。尽管如此，我们证明：通过将样本中倒数第二次成功设为阈值，并停止在该阈值之后首次观察到的成功上，可以保证获得$1/4$的获胜概率。