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「What are the odds?」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

An unusual lottery result made the news recently: on October 1, 2022, the PCSO Grand Lotto in the Philippines, which draws six numbers from to at random, managed to draw the numbers (though the balls were actually drawn in the order ). In other words, they drew exactly six multiples of nine from to . In addition, a total of tickets were bought with this winning combination, whose owners then had to split the million peso jackpot (about million USD) among themselves. This rais

Whenever an event like this happens, journalists often contact mathematicians to ask the question: “What are the odds of this happening?”, and in fact I myself received one such inquiry this time around. This is a number that is not too difficult to compute – in this case, the probability of the lottery producing the six numbers in some order turn out to be in – and such a number is often dutifully provided to such journalists, who in turn report it as some sort of quantitati

已知结果和反例

But on the previous draw of the same lottery, on September 28, 2022, the unremarkable sequence of numbers were drawn (again in a different order), and no tickets ended up claiming the jackpot. The probability of the lottery producing the six numbers is also in – just as likely or as unlikely as the October 1 numbers . Indeed, the whole point of drawing the numbers randomly is to make each of the possible outcomes (whether they be “unusual” or “unremarkable”) equally likely. S

Part of the explanation surely lies in the unusually large number ( ) of lottery winners on October 1, but I will set that aspect of the story aside until the end of this post. The more general points that I want to make with these sorts of situations are:

证明或构造的主线

To explain these points it is convenient to adopt the framework of Bayesian probability . In this framework, one imagines that there are competing hypotheses to explain the world, and that one assigns a probability to each such hypothesis representing one’s belief in the truth of that hypothesis. For simplicity, let us assume that there are just two competing hypotheses to be entertained: the null hypothesis , and an alternative hypothesis . For instance, in our lottery examp

At any given point in time, a person would have a probability assigned to the null hypothesis, and a probability assigned to the alternative hypothesis; in this simplified model where there are only two hypotheses under consideration, these probabilities must add to one, but of course if there were additional hypotheses beyond these two then this would no longer be the case.

阅读时建议盯住的点

Bayesian probability does not provide a rule for calculating the initial (or prior ) probabilities , that one starts with; these may depend on the subjective experiences and biases of the person considering the hypothesis. For instance, one person might have quite a bit of prior faith in the lottery system, and assign the probabilities and . Another person might have quite a bit of prior cynicism, and perhaps assign and . One cannot use purely mathematical arguments to determ

What Bayesian probability does do, however, is provide a rule to update these probabilities , in view of new information to provide posterior probabilities , . In our example, the new information would be the fact that the October 1 lottery numbers were (in some order). The update is given by the famous Bayes theorem

值得单独记下的条目

• The question “what are the odds of happening” is often easy to answer mathematically, but it is not the correct question to ask.
• The question “what is the probability that an alternative hypothesis is the truth” is (one of) the correct questions to ask, but is very difficult to answer (it involves both mathematical and non-mathematical considerations).
• The answer to the first question is one of the quantities needed to calculate the answer to the second, but it is far from the only such quantity. Most of the other quantities involved cannot be calculated exactly.
• However, by making some educated guesses, one can still sometimes get a very rough gauge of which events are “more surprising” than others, in that they would lead to relatively higher answers to the second question.
• Null hypothesis : The lottery is run in a completely fair and random fashion.
• Alternative hypothesis : The lottery is rigged by some corrupt officials for their personal gain.
• The prior odds of the alternative hypothesis;
• The probability that the event occurs under the null hypothesis ; and

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