研究表明，拋硬幣的機率不是 50-50

拋硬幣的行為是一個古老的概念。它是做出艱難決定的一種非常簡單的方法，因為它為任一結果提供了均等的機會。比例一直是50：50。這個假設是公平的，因為所有硬幣都有兩面，並且當有人翻轉硬幣時，硬幣出現在任何一面的機會均等。然而，美國數學家 Persi Diaconis 進行的一項研究表明，拋硬幣的機率在過去並不是 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.

根據美國數學家 Persi Diaconis 不久前進行的一項研究，拋硬幣的機率不是 50:50。他的研究表明，一枚硬幣落地時落在同一面的可能性更高，這種現像被稱為「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."

阿姆斯特丹大學研究心理學方法的博士生 Frantisek Bartos 進行的預印本研究以 Diaconis 的原始論文為基礎。他的結果與戴康尼斯的結果一致。他在X（以前的Twitter）上分享道：「我們發現了壓倒性的證據，證明Diaconis 及其同事在2007 年預測的『同方』偏見：如果你開始單挑，硬幣更有可能正面落地，反之亦然. 在我們的樣本中，平均估計值為 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.

這種機率模型被稱為“戴科尼斯模型”，它改變了人類長期以來對拋硬幣的理解方式。根據 IFL Science 報道，另一個團隊談到了該模型，「根據 Diaconis 模型，進動導致硬幣在空中停留更多時間，且初始面朝上。因此，硬幣有更高的機會落在同一面。」開始時的一側（即“同側偏差”）。該團隊付出了巨大的努力，讓 48 個人拋擲了來自 46 個不同國家的 350,757 枚硬幣，得出了他們的結果。

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.

結果發現，硬幣有 51% 的機會落在拋擲的同一面，這與巴托斯得到的結果相同。此外，研究團隊還發現，拋硬幣的機率受到拋硬幣的個人的影響。有些人表現出偏向某一方，而其他許多人則沒有這種偏見。研究小組的結論是，拋硬幣行為受到拋硬幣者的微妙影響。

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

雖然這些數字看起來並不大，但在某些情況下它們可能會帶來可預測的結果。

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.

巴托斯提供了一個例子，他說：「觀察到的偏差的大小可以透過投注場景來說明。如果你在拋硬幣的結果上賭一美元（即支付1 美元進入並贏得0 或2 美元，具體取決於）並重複下注 1,000 次，並且知道拋硬幣的起始位置，平均可以為您贏得 19 美元。他們接著解釋了二十一點遊戲中的玩法。

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.

他們表示，這將超過賭場在六副牌的二十一點遊戲中與具有最佳策略的玩家相比所具有的優勢。然後，該團隊揭示了賭場如何透過類似的投注賺取 5 美元，但這將小於他們在單零輪盤賭中的優勢，在單零輪盤賭中，他們平均賺取 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."

閱讀他們的研究的人自然會好奇他們的結果將如何影響傳統的拋硬幣。他們回應這句話時說：“當用拋硬幣進行高風險決策時，最好隱藏硬幣的起始位置。”

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

您可以在 Twitter 上關注@BartosFra，以了解更多有趣的統計數據。

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

編者註：本文原刊於 2023 年 10 月 13 日。