数学2024researchgate.net
公平的硬币往往落在起始朝上的一面：来自 350757 次抛硬币的证据
Fair coins tend to land on the side they started (2023)
进阶 · 约 45 分钟读完前置知识：概率论基础、统计推断、贝叶斯回归基础
导读
这篇论文基于 350,757 次真实抛硬币实验，验证了 Diaconis 等人 2007 年的理论预测：由于人手抛掷的物理不完美性，硬币更倾向于落在开始时朝上的同一面，实验测得同面偏差约为 50.8%。作者指出若事先知道起始面并押注同面，长期可获得微小优势。
HN 关注点：这一反直觉结论是否可信，讨论焦点集中在样本量、抛掷者与作者身份重合可能引入的偏差，以及第一作者亲自下场回应质疑。
HN 评论精选
这几条评论里：质疑 1 补充 2 亲历 1
质疑质疑有效样本量过小且抛掷者即作者，可能无意中习得技巧造成偏差
论文看似样本量很大，但实际抛掷者只有 48 人。部分测试者的视频显示抛掷转速极低、只有 1-2 次翻转，而他们又各自抛了数千次，方式可能相同。所以有效样本其实很小（N=48），抛掷不规范的测试者会对结果产生不成比例的影响。作者背景似乎也不以统计学见长，48 位作者中除 3 人外都参与抛硬币，有些人可能更像受试者而非作者。自己当自己研究的受试者难道不会引入偏差吗？如果你想证明硬币会偏向某一侧，你自然会练出相应技巧，尤其是低速低抛时——从结果看一些抛掷者显然学得很好。如果作者们原本相信的是相反结论，我敢说他们同样能收集数据写出一篇'证明'相反结果的论文。
acyou查看英文原文
The paper looks like it has a large sample size, but it actually has a sample size of only 48 testers/flippers. Some of the videos of those testers show very low, low-rpm coin tosses, we're talking only 1-2 flips. Where they also flipped thousands of times, presumably in the same way. So there is actually a very small sample size in the study (N = 48), where testers that don't flip properly (low rpm, low height, few coin rotations) can affect the results disproportionately. Doesn't look like the study author backgrounds are particularly focused on statistics. I would presume with 48 authors (all but 3 of which flipped coins for the study), the role of some might have been more test subject than author. And isn't being the subject in your own study going to introduce some bias? Surely if you're trying to prove to yourself that the coins land on one side or another given some factor, you will learn the technique to do it, especially if you are doing a low-rpm, low flip. Based on the study results, some of the flippers appear to have learned this quite well. If the flippers (authors) had been convinced of the opposite (fair coins tend to land on the opposite side from which they started) and done the same study, I bet they could have collected data and written a paper with the results proving that outcome.
亲历第一作者回应质疑：视频是审计用途、机器抛掷违背研究目的、无经费且署名合理
我是论文第一作者，来澄清几个问题：低转速抛掷是因为大部分录像用的是约 30FPS 的廉价摄像头，硬币转速超过传感器采样率，看起来像没转。拍视频是为了核查数据收集和审计结果。关于造抛币机器：本研究关注的是人类抛硬币，DHM (2007) 理论认为正是人手抛掷的不完美导致同面偏差，用机器就违背了实验目的。关于作者多和浪费公款：实验是业余时间做的，没有经费，没浪费钱；贡献整天抛硬币的学生获得署名是应得的，没有他们实验无法完成。关于不规范抛掷：我们要求大家像解决赌注一样抛硬币且至少翻转一次，多数人的偏差随时间递减……
fbartos查看英文原文
Hi, I'm the first author of the manuscript, so I thought I could answer some of the questions and clarify some issues (all details are in the manuscript, but who has the time to read it ;) Low RPM tosses: Most of the recordings are on crapy webcams with ~ 30FPS. The coin spin usually much faster than the sensor can record which results in often non-spinning-looking flips. Why did we take the videos in the first place? To check that everyone collected the data and to audit the results. Building a flipping matching: The study is concerned with human coin flips. Diaconis, Holmes, and Montgomery's (DHM, 2007) paper theorize that the imperfection of human flips causes the same-side bias. Building a machine completely defeats the purpose of the experiment. Many authors and wasted public funding: We did the experiment in our free time and we had no funding for the study = no money was wasted. Also, I don't understand why are so many people angry that students who contributed their free time and spent the whole day flipping coins with us were rewarded with co-authorship. The experiment would be impossible to do without them. Improper tosses: Not everyone flips coin perfectly and some people are much worse at flipping than others. We instructed everyone to flip the coin as if they were to settle a bet and that the coin has to flip at least once (at least one flip would create bias for the opposite side). We find that for most people, the bias decreased over time which suggests that people might get better at flipping by practice = decrease the bias and it also discredits the theory that they learned how to be biased on purpose. From my own experience - I flipped coins more than 20,000 times and I have no clue how to bias it. Also, we did a couple of sensitivity analyses excluding outliers - the effect decreased a bit but we still found plentiful evidence for DHM. If you doubt my stats background, you are more than welcome to re-analyze the data on your own. They are available on OSF: https://osf.io/mhvp7/ (including cleaning scripts etc). Frantisek Bartos
补充介绍 von Neumann 去偏技巧：成对抛掷取相反结果可消除偏差
从不公平的抛硬币中得到公平结果的简单方法：连续抛两次，且两次都以同一面朝上开始，这样两次抛掷的不公平程度相同。如果得到正正或反反就作废重来，直到得到正反或反正——这两种结果概率相等（比如 HT 记为'正面'，TH 记为'反面'）。即使硬币 99% 的时间落正面，只要偏差一致，这个方法也有效（只是可能要多抛几次）。
NameError查看英文原文
Easy way to get a fair result from an unfair coin toss: Flip the coin twice in a row, in this case starting with the same side facing up both times, so it's equally unfair for both tosses. If you get heads-heads or tails-tails, discard and start over until you get either heads-tails or tails-heads, which have equal probabilities (so you can say something like HT = "heads" and TH = "tails"). This works even if the coin lands heads 99% of the time, as long as it's consistent (but you'll probably have to flip a bunch of times in that case).
补充以玄学式类比解释现象（信息量低，可跳过）
这显然是'现实守恒定律'在起作用。同样地，当你第一次听到一个词，接下来会突然连续听到五次；见过某人一次后就会到处遇到他。这是因为重复过去的现实比创造新的现实更'便宜'。
swayvil查看英文原文
This is clearly the law of conservation of reality at work. Likewise, when you hear a word for the first time suddenly you hear it five times in a row. Or if you see somebody once you suddenly start running into them all over the place. It's because it's cheaper to repeat past realities than to create new ones.
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