陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Small samples, and the margin of error」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

The U.S. presidential election is now only a few weeks away. The politics of this election are of course interesting and important, but I do not want to discuss these topics here (there is not exactly a shortage of other venues for such a discussion), and would request that readers refrain from doing so in the comments to this post. However, 一个常见想法是 it would be apropos to talk about some of the basic mathematics underlying electoral polling, and specifically to explain the fa

Take for instance a nationwide poll of U.S. voters on which presidential candidate they intend to vote for. A typical poll will ask a number of randomly selected voters for their opinion; a typical value here is . In contrast, the total voting-eligible population of the U.S. – let’s call this set – is about 200 million. (The actual turnout in the election is likely to be closer to 100 million, but let’s ignore this fact for the sake of discussion.) Thus, such a poll would sam

已知结果和反例

I’ll give a rigorous proof of a weaker version of the above statement (giving a margin of error of about 7%, rather than 3%) in an appendix at the end of this post. But the main point of my post here is a little different, namely to address the common misconception that the accuracy of a poll is a function of the relative sample size rather than the absolute sample size, which would suggest that a poll involving only 0.0005% of the population could not possibly have a margin

Not all polls are created equal; there are a certain number of hypotheses on the methodology and effectiveness of the poll that we have to assume in order to make our mathematical conclusions valid. We will make the following idealised assumptions:

证明或构造的主线

Polls which deviate significantly from these hypotheses (e.g. due to complex questions, self-selection or other selection bias, confirmation bias, inaccurate responses, a high refusal rate, variable poll size, or clustering) will generally be less accurate than an idealised poll with the same sample size. Of course, there is a substantial literature in statistics (and polling methodology) devoted to measuring, mitigating, avoiding, or compensating for these less ideal situati

As stated in the introduction, we let be the proportion of the entire population that will vote for , and be the proportion of the polled voters that will vote for (which, by Hypotheses 2 and 3, is exactly equal to the proportion of polled voters that say that they will vote for ). Under the above idealised conditions, if the number of polled voters is 1,000, and the size of the population is 200 million, then the margin of error is about 3%, thus . (See this margin of error

阅读时建议盯住的点

There is an important subtlety here: it is only the unconditional probability of the event that is guaranteed to be greater than 0.95. If one has additional prior information about and , then the conditional probability of this event, relative to this information, may be very different. For instance, if one had, prior to the poll, a very good reason to believe that is almost certainly between 0.4 and 0.6, and then the poll reports to be 0.1, then the conditional probability t

One special case of the above point is worth emphasising: the statement that is true with at least 95% probability is only valid before one actually conducts the poll and finds out the value of . Once is computed, the statement is either true or false, i.e. occurs with probability 1 or 0 (unless one takes a Bayesian approach, as mentioned above). [This phenomenon of course occurs all the time in probability. For instance, if x denotes the outcome of rolling a fair six-sided d

值得单独记下的条目

• Fixed poll size. The number of polled voters is fixed in advance; in particular, one cannot keep polling until one has achieved some desired outcome, and then stop.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源