Live win-probability forecasts are now ubiquitous in sports broadcasts, and retrospective commentary often cites the largest win probability attained by a team that ultimately loses as evidence of a "collapse." Interpreting such extrema requires a reference distribution under correct specification. Modeling the forecast sequence as the Doob martingale of conditional win probabilities for a binary terminal outcome, we derive sharp distributional laws for its path maximum, including the conditional law given an eventual loss. In discrete time, we quantify explicit correction terms (last-step crossings and overshoots); under continuous-path regularity these corrections disappear, yielding exact identities. We further obtain closed-form distributions for two extensions: the maximal win probability attained by the eventual loser in a two-player game and the minimal win probability attained by the eventual winner in an n-player game. The resulting formulas furnish practical benchmarks for diagnosing sequential forecast calibration.

翻译：实时胜率预测如今在体育转播中已无处不在，而回顾性评论常引用最终输球一方曾达到的最大胜率作为“崩盘”的证据。解释此类极值需要在正确设定下建立参考分布。通过将预测序列建模为二元终局结果条件胜率的Doob鞅，我们推导了其路径最大值的尖锐分布律，包括给定最终输球条件下的条件律。在离散时间情形中，我们量化了显式修正项（最后一步穿越与超调）；在连续路径正则性条件下这些修正项消失，得到精确恒等式。我们进一步获得两个扩展问题的闭式分布：双人游戏中最终输家曾达到的最大胜率，以及n人游戏中最终赢家曾达到的最小胜率。所得公式为诊断序列预测校准提供了实用基准。