研究表明，抛硬币的概率不是 50-50

抛硬币的行为是一个古老的概念。它是做出艰难决定的一种非常简单的方法，因为它为任一结果提供了均等的机会。比例一直是50：50。这个假设是公平的，因为所有硬币都有两面，并且当有人翻转硬币时，硬币出现在任何一面的机会均等。然而，美国数学家 Persi Diaconis 进行的一项研究表明，抛硬币的概率在过去并不是 50:50。

Coin tosses are not a 50-50 probability, according to a study conducted by American mathematician Persi Diaconis sometime back. His study revealed that a coin has a higher chance of landing on the same side as it started, a phenomenon known as the "same-side" bias.

根据美国数学家 Persi Diaconis 不久前进行的一项研究，抛硬币的概率不是 50:50。他的研究表明，一枚硬币落地时落在同一面的可能性更高，这种现象被称为“同面”偏差。

A pre-print study conducted by Frantisek Bartos, a PhD candidate studying psychological methods at the University of Amsterdam, built off the original paper from Diaconis. His results aligned with that of Diaconis. He shared on X (formerly Twitter): "We found overwhelming evidence for a 'same-side' bias predicted by Diaconis and colleagues in 2007: If you start heads-up, the coin is more likely to land heads-up and vice versa. How large is the bias? In our sample, the mean estimate is 50.8%, CI."

阿姆斯特丹大学研究心理学方法的博士生 Frantisek Bartos 进行的预印本研究以 Diaconis 的原始论文为基础。他的结果与戴康尼斯的结果一致。他在 X（以前的 Twitter）上分享道：“我们发现了压倒性的证据，证明 Diaconis 及其同事在 2007 年预测的‘同方’偏见：如果你开始单挑，硬币更有可能正面落地，反之亦然. 在我们的样本中，平均估计值为 50.8%，CI。”

The probability model, called the "Diaconis Model," changes the way humans have been understanding coin tosses for a long time. According to IFL Science, another team spoke about the model, "According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started (i.e., 'same-side bias')." The team took a herculean effort and got 48 people to flip 350,757 coins from 46 different countries to come up with their results.

这种概率模型被称为“戴科尼斯模型”，它改变了人类长期以来对抛硬币的理解方式。据 IFL Science 报道，另一个团队谈到了该模型，“根据 Diaconis 模型，进动导致硬币在空中停留更多时间，且初始面朝上。因此，硬币有更高的机会落在同一面。”开始时的一侧（即“同侧偏差”）。”该团队付出了巨大的努力，让 48 个人抛掷了来自 46 个不同国家的 350,757 枚硬币，得出了他们的结果。

It was found that coins had a 51% chance to land on the same side they were tossed from, the same results Bartos got. Additionally, the team also discovered that the probability of coin tosses was affected by the individual tossing it. Some were shown to favor a certain side, while many others had no such bias. The team came to the conclusion that coin tosses were subtly influenced by the person doing it.

结果发现，硬币有 51% 的机会落在抛掷的同一面，这与巴托斯得到的结果相同。此外，研究小组还发现，抛硬币的概率受到抛硬币的个人的影响。有些人表现出偏向某一方，而其他许多人则没有这种偏见。研究小组得出的结论是，抛硬币行为受到抛硬币者的微妙影响。

While these numbers may not seem huge, they could lead to predictable results in certain scenarios.

虽然这些数字可能看起来并不大，但在某些情况下它们可能会带来可预测的结果。

Bartos provided an example, saying, "The magnitude of the observed bias can be illustrated using a betting scenario. If you bet a dollar on the outcome of a coin toss (i.e., paying 1 dollar to enter and winning either 0 or 2 dollars depending on the outcome) and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average." They went on to explain how it would play out in a game of blackjack.

巴托斯提供了一个例子，他说：“观察到的偏差的大小可以通过投注场景来说明。如果你在抛硬币的结果上赌一美元（即支付 1 美元进入并赢得 0 或 2 美元，具体取决于）并重复下注 1,000 次，知道抛硬币的起始位置平均可以为您赢得 19 美元。”他们接着解释了二十一点游戏中的玩法。

They state that this would be more than the advantage that a casino had for a game of blackjack with six decks against a player with optimal strategy. The team then reveals how the casino would make five dollars on a comparable bet, but it would be less than the advantage they had in single-zero roulette, where they would make 27 dollars on average.

他们表示，这将超过赌场在六副牌的二十一点游戏中与具有最佳策略的玩家对抗的优势。然后，该团队揭示了赌场如何通过类似的投注赚取 5 美元，但这将小于他们在单零轮盘赌中的优势，在单零轮盘赌中，他们平均赚取 27 美元。

People who read their study would naturally be curious about how their results would affect a conventional coin toss. They reply to this saying, "When coin flips are used for high-stakes decision-making, the starting position of the coin is best concealed."

阅读他们的研究的人自然会好奇他们的结果将如何影响传统的抛硬币。他们回应这样的说法：“当用抛硬币进行高风险决策时，最好隐藏硬币的起始位置。”

You can follow @BartosFra on Twitter for more interesting takes on statistics.

您可以在 Twitter 上关注@BartosFra，了解更多有趣的统计数据。

Editor's note: This article was originally published on October 13, 2023. It has since been updated.

编者注：本文最初发表于 2023 年 10 月 13 日。此后已更新。