Concise Summary简洁概述
Yudkowsky uses the lottery as a precise mathematical model for a universal epistemic principle: every hypothesis occupies a space of possibilities, and the evidence required to single it out must be at least as strong as that space is large. He introduces likelihood ratios and information-theoretic bits to quantify this, showing that 131 million lottery tickets demand roughly 34 bits of evidence for confident belief in a winner. The broader lesson is that you cannot bypass the math by willpower or wishful thinking — insufficient evidence produces inaccurate beliefs, exactly as surely as an empty fuel tank stops a car.
Yudkowsky 用彩票作为一个精确的数学模型，来阐明一条普遍的认识论原则：每个假设都占据一个可能性空间，而把它从中单独挑出所需的证据，必须至少和那个空间一样强大。他引入似然比和信息论中的「比特」来量化这一点，并证明：在 1.31 亿张彩票中找到赢家，大约需要 34 比特的证据。更宏观的教训是：你无法用意志力或一厢情愿来绕过这道数学——证据不足，产生的就是不准确的信念，就像油箱空了就是无法开车一样确定。
Infographic信息图
Evidence must outweigh the possibility space
证据必须能压过可能性空间
A filter as strong as the number of alternatives only gives you 50/50 odds. You need to go further to reach confident belief.
一个和备选方案数量一样强的过滤器，只能给你 50/50 的胜算。要达到有把握的信念，你必须走得更远。
Likelihood ratios multiply; bits add
似然比相乘；比特相加
Each independent test multiplies the odds ratio, which is the same as adding bits of evidence — a clean, composable arithmetic of belief.
每次独立测试都会乘以赔率比，这等同于累加证据的比特数——一种简洁、可组合的信念算术。
The fuel tank analogy
油箱类比
Believing without adequate evidence is like driving without fuel — you can pretend the car is moving, but accurate arrival requires real fuel.
没有充足证据就相信，就像没有燃料就开车——你可以假装车在动，但准确到达目的地需要真实的燃料。
Prior probability sets the starting line
先验概率决定起点
The rarer a hypothesis a priori, the more evidence is needed. A 1-in-130-million prior demands far more bits than a coin flip.
一个假设的先验概率越低，所需的证据就越多。百分之一百三十分之一的先验，远比抛硬币需要更多比特。
You cannot exempt yourself from the math
你无法豁免于这道数学
Stopping at the first promising-looking result and ignoring Bayesian rules still leaves, on average, 131 false positives for every true winner.
停在第一个看起来有希望的结果上、无视贝叶斯规则，平均每找到一个真赢家仍会留下 131 个误报。
Detailed Summary详细概述
The Lottery as an Epistemic Model
Yudkowsky opens with a seemingly narrow puzzle: what would it take to win the lottery through sheer evidence rather than luck? With 131,115,985 possible tickets, a randomly chosen ticket has roughly a 1-in-131-million chance of winning. To believe a specific ticket is the winner, you need evidence selective enough to distinguish it from all 131,115,984 losers.
Likelihood Ratios and the Multiplication of Evidence
He introduces a hypothetical black box: it beeps with certainty for a winning ticket and with 25% probability for any losing ticket — a likelihood ratio of 4:1. Using such a box doesn't win you the lottery outright; it reduces your losing pool from 131 million to about 33 million. Two independent boxes compound to 16:1, cutting the losers to ~8 million. The key insight is that evidence multiplies as odds ratios — each independent test multiplies the existing odds by the new likelihood ratio.
Measuring Evidence in Bits
Yudkowsky introduces mathematician's bits: the log base-1/2 of probabilities. A 4:1 likelihood ratio equals 2 bits of evidence. Since 131,115,984 ≈ 2^27, you might think 27 bits is enough — but that only yields 50/50 odds, leaving on average one loser alongside the winner. To get odds of 128:1 (less than 1% error), you need around 34 bits — roughly 17 independent 4:1 tests.
The General Principle
The lottery is a vehicle for a universal rule:
- The larger the possibility space, the more evidence you need.
- The more a priori unlikely the hypothesis compared to alternatives, the more you need.
- The higher the confidence required, the more you need.
These factors combine multiplicatively in odds space (additively in bits).
You Cannot Opt Out
Yudkowsky is emphatic: these are not bureaucratic Bayesian rules you can dismiss as "ivory-tower." If you stop at the first combination that passes 10 independent 1-in-a-million tests, you'll still have on average 131 false positives. Jumping to a strong conclusion on inadequate evidence is a mathematical error, not a stylistic one.
"You cannot form accurate beliefs based on inadequate evidence."
The Fuel Tank Analogy
The essay closes with a vivid analogy: believing on inadequate evidence is like trying to drive without fuel. You can shut your eyes and pretend the car moves. But arriving accurately — at true beliefs about the world — requires evidence-fuel, and the further you want to go (the more specific the belief, the lower the prior, the higher the required confidence), the more fuel you need. The laws governing this are not optional.
彩票作为认识论模型
Yudkowsky 以一个看似狭窄的难题开篇：纯靠证据而非运气来赢得彩票，需要什么？在 131,115,985 种可能的组合里，随机选一张票中奖的概率大约是 1.31 亿分之一。要相信某张具体的票是赢家，你需要的证据必须具有足够的筛选力，能把它从所有 131,115,984 张输家中区分出来。
似然比与证据的乘法
他引入了一个假想的黑匣子：对赢家组合必然鸣叫，对任何输家组合以 25% 的概率鸣叫——似然比为 4:1。使用这样一个盒子并不能直接赢得彩票；它只是把你的输家池从 1.31 亿缩减到大约 3300 万。两个独立的盒子累积到 16:1，把输家进一步削减到约 800 万。关键洞见在于：证据以赔率之比的形式相乘——每次独立测试都把现有赔率乘以新的似然比。
用比特衡量证据
Yudkowsky 引入了数学家的「比特」：以 1/2 为底的概率对数。4:1 的似然比等于 2 比特的证据。由于 131,115,984 ≈ 2^27，你可能以为 27 比特就够了——但那只能产生 50/50 的赔率，平均仍有一张输家和赢家并列。要达到 128:1 的赔率（出错率低于 1%），大约需要 34 比特——大致相当于 17 次独立的 4:1 测试。
普遍原则
彩票是传递一条普遍规则的载体：
- 可能性空间越大，所需证据越多。
- 假设相对于备选方案的先验概率越低，所需越多。
- 要求的置信度越高，所需越多。
这些因素在赔率空间中以乘法方式叠加（在比特空间中则以加法方式）。
你无法选择退出
Yudkowsky 态度鲜明：这些不是你可以斥为「象牙塔」而置之不理的官僚式贝叶斯规则。如果你停在第一个通过了 10 次百万分之一独立测试的组合上，你仍然平均会剩下 131 个误报。凭不充分的证据跳向强结论，是一个数学错误，而非风格问题。
「你不能在证据不足的情况下形成准确的信念。」
油箱类比
文章以一个生动的类比收尾：凭不充分的证据相信，就像试图在没有燃料的情况下开车。你可以闭上眼睛假装车在行驶。但准确到达目的地——对世界形成真实的信念——需要证据-燃料，你想走得越远（信念越具体、先验越低、要求的置信度越高），就需要越多的燃料。支配这一切的规律不是可选项。
FAQ常见问答
Why isn't evidence equal to the size of the possibility space enough to win the lottery?为什么与可能性空间大小相等的证据不足以赢得彩票？
Evidence as strong as 131 million to one only yields even odds — one loser survives alongside the winner on average. To get, say, 99% confidence you need to supply roughly 1% of 131 million worth of extra evidence, i.e., you must overshoot the space by a factor equal to your required confidence margin.
强度达到 1.31 亿对一的证据，只能产生均等的赔率——平均仍有一张输家和赢家并列。要达到 99% 的置信度，你需要再提供大约 1.31 亿的 1% 那么多的额外证据，也就是说，你必须超出这个空间一个与你所要求的置信度边际相等的倍数。
What exactly is a "likelihood ratio" and why does it matter?「似然比」究竟是什么，为什么重要？
A likelihood ratio compares how probable the evidence is given the hypothesis is true, versus given it is false. A 4:1 ratio means the evidence is 4× more likely under the true hypothesis. Bayesian updating works by multiplying your prior odds by the likelihood ratio, so the ratio directly measures how much each piece of evidence moves the needle.
似然比比较的是：假设假设为真时证据出现的概率，与假设假设为假时的概率之比。4:1 的似然比意味着该证据在真实假设下出现的可能性是假的 4 倍。贝叶斯更新的工作方式是将先验赔率乘以似然比，因此似然比直接衡量了每条证据让指针移动多少。
Why measure evidence in "bits" rather than raw probabilities?为什么用「比特」而不是原始概率来衡量证据？
Bits are additive while probabilities multiply — adding bits of evidence is arithmetically simpler and maps onto information theory. Two independent 2-bit tests give 4 bits total, whereas the equivalent probability operation requires multiplying 1/4 × 1/4 = 1/16 and then interpreting the result. Bits also make the analogy to information storage intuitive.
比特是可加的，而概率需要相乘——加比特在算术上更简洁，并与信息论接轨。两次独立的 2 比特测试总共给出 4 比特，而等价的概率运算需要把 1/4 × 1/4 = 1/16 乘出来再加以解读。比特还让与信息存储的类比变得直观。
Does this principle apply outside lotteries — to everyday beliefs and scientific claims?这条原则在彩票之外——在日常信念和科学主张中——也适用吗？
Yes, and that's Yudkowsky's point. Any belief about a large space of possibilities — a medical diagnosis among thousands of conditions, a conspiracy theory among countless alternatives, a scientific hypothesis — requires evidence proportional to the prior improbability. The lottery merely makes the math explicit and unambiguous.
是的，这正是 Yudkowsky 的论点。任何关于一个大型可能性空间的信念——在数千种病症中做出医学诊断、在无数备选方案中支持一个阴谋论、提出一个科学假说——都需要与先验不可能性成比例的证据。彩票只是让数学变得明确而无歧义。
What does the essay mean that you "cannot form accurate beliefs based on inadequate evidence"?文章中「你不能在证据不足的情况下形成准确的信念」是什么意思？
It means that epistemic accuracy is not under voluntary control. You can choose to believe on insufficient evidence, but you cannot choose to be right while doing so at the same rate as someone with sufficient evidence. The connection between evidence and accuracy is mathematical, not social — like the connection between fuel and distance traveled.
这意味着认识论上的准确性不在你的意志控制之下。你可以选择凭不充分的证据相信，但你无法在这样做的同时选择以与拥有充分证据的人相同的频率正确。证据与准确性之间的联系是数学性的，而非社会性的——就像燃料与行驶距离之间的联系一样。
How does this relate to the broader Sequences on rationality?这与更宏观的「理性序列」有何关联？
This essay provides the quantitative backbone for Yudkowsky's rationality project. Earlier essays establish that beliefs should be "entangled" with reality; this one specifies how much entanglement is required and why. It sets up the machinery (Bayes, likelihood ratios, bits) that the rest of the Sequences use to diagnose and fix specific cognitive errors.
这篇文章为 Yudkowsky 的理性项目提供了定量骨干。之前的文章确立了信念应当与现实「纠缠」；这篇则明确说明需要多少纠缠以及原因。它建立了后续序列用来诊断和修正特定认知错误的机制（贝叶斯、似然比、比特）。
In-depth Analysis · Pros & Cons深入解读 · 优缺点
This essay is the Sequences' pivot from qualitative to quantitative epistemology: having established what evidence is and why beliefs must be grounded in it, Yudkowsky now shows how much is required. The lottery device is pedagogically elegant, converting an abstract principle into arithmetic anyone can verify.
这篇文章是序列从定性认识论转向定量认识论的关键节点：在确立了证据是什么以及信念为何必须以证据为基础之后，Yudkowsky 现在展示需要多少。彩票装置在教学上极为优雅，将一条抽象原则转化为任何人都能验证的算术。
- Concreteness makes the abstract inevitable具体性使抽象原则变得不可逃避By anchoring the argument in a lottery with exact, countable possibilities, Yudkowsky ensures the reader cannot escape into vagueness — the math is right there, checkable.通过将论证锚定在一个具有精确可数可能性的彩票上，Yudkowsky 确保读者无法逃入模糊性——数学就在那里，可以验证。
- Likelihood ratios as a portable tool似然比作为便携工具Introducing the likelihood ratio concept cleanly sets up Bayesian updating as a practical skill, not just a theorem — readers leave with a mental calculator they can apply to real claims.干净地引入似然比概念，将贝叶斯更新建立为一种实践技能，而不仅仅是一个定理——读者离开时拥有了一个可以应用于真实主张的心理计算器。
- The overshoot insight is non-obvious and important「超出」洞见是非显然且重要的The point that evidence equal to the space size only gives 50/50 odds is a genuine surprise that corrects a natural intuition, making the essay memorable and instructive.与空间大小相等的证据只能给出 50/50 赔率这一点，是一个真正令人惊讶的洞见，它纠正了一种自然直觉，使文章令人难忘且富有启发性。
- The fuel analogy caps the argument with rhetorical force油箱类比以修辞力量收尾The car-without-fuel image is both memorable and accurate: it conveys that the constraint is physical, not merely logical, and that pretending otherwise doesn't change outcomes.没有燃料的汽车意象既令人难忘又准确：它传达了这个约束是物理性的，而非仅仅是逻辑性的，假装不存在并不会改变结果。
- The lottery is an unusually clean case彩票是一个异常干净的案例Real-world hypothesis spaces are not enumerable, priors are contested, and likelihood ratios are rarely known. The essay establishes the principle soundly but doesn't grapple with the hard problem of estimating these quantities in practice, which is where most epistemic difficulty lies.现实世界的假设空间不可枚举，先验存在争议，似然比也很少是已知的。文章确立了这条原则，但没有处理实践中估算这些量的困难问题——而这恰恰是大多数认识论困难所在。
- Independence assumption is quietly load-bearing独立性假设是悄悄承重的The arithmetic that lets bits add requires each test to be truly independent. In real epistemic practice, evidence sources are often correlated (same underlying data, same methodological bias), which can drastically reduce the effective evidence. This caveat is unaddressed.让比特相加的算术要求每次测试都是真正独立的。在真实的认识论实践中，证据来源往往是相关的（相同的底层数据、相同的方法论偏差），这可能会大幅降低有效证据。这一警告未被提及。
- Confidence thresholds are presented as arbitrary置信度阈值被呈现为任意的The "less than 1% probability of being wrong" threshold is described as "arbitrarily defined" but then treated as if choosing it is unproblematic. In reality, the appropriate threshold depends on stakes, costs of errors, and base rates — decisions the essay leaves entirely to the reader.「出错概率低于 1%」的阈值被描述为「任意定义的」，但随后被当作选择它没有问题来处理。实际上，合适的阈值取决于赌注、错误成本和基础率——文章将这些决定完全留给读者。
- The approach proves more than applies to most human beliefs这种方法证明的范围超出了对大多数人类信念的适用性The framework works perfectly for finitely enumerable spaces like lotteries. For beliefs about open-ended causal claims — "will this policy reduce crime?" — no one can enumerate the hypothesis space or assign principled likelihood ratios, which limits the framework's direct applicability to everyday reasoning.这个框架对于像彩票这样有限可枚举的空间完美有效。但对于开放式因果主张的信念——「这项政策会减少犯罪吗？」——没有人能枚举假设空间或分配有原则的似然比，这限制了该框架对日常推理的直接适用性。
A clean, mathematically honest treatment of a principle most people violate constantly without realizing it. The lottery conceit is brilliant pedagogy, and the overshoot insight alone is worth the read. Its limitation is that it does the easy half of Bayesian epistemology — proving the rule exists — while leaving untouched the genuinely hard half: estimating priors and likelihood ratios in messy real-world situations.
对一条大多数人在不知不觉中不断违反的原则，做出了清晰、数学上诚实的阐述。彩票构思是绝妙的教学法，单是「超出」洞见就值得一读。它的局限在于：它完成了贝叶斯认识论容易的那一半——证明规则的存在——而完全未触及真正困难的那一半：在混乱的现实情境中估算先验和似然比。
Original Text原文
Previously, I defined evidence as “an event entangled, by links of cause and effect, with whatever you want to know about,” and entangled as “happening differently for different possible states of the target.” So how much entanglement—how much rational evidence—is required to support a belief?
Let’s start with a question simple enough to be mathematical: How hard would you have to entangle yourself with the lottery in order to win? Suppose there are seventy balls, drawn without replacement, and six numbers to match for the win. Then there are 131,115,985 possible winning combinations, hence a randomly selected ticket would have a 1/131,115,985 probability of winning (0.0000007%). To win the lottery, you would need evidence selective enough to visibly favor one combination over 131,115,984 alternatives.
Suppose there are some tests you can perform which discriminate, probabilistically, between winning and losing lottery numbers. For example, you can punch a combination into a little black box that always beeps if the combination is the winner, and has only a 1/4 (25%) chance of beeping if the combination is wrong. In Bayesian terms, we would say the likelihood ratio is 4 to 1. This means that the box is 4 times as likely to beep when we punch in a correct combination, compared to how likely it is to beep for an incorrect combination.
There are still a whole lot of possible combinations. If you punch in 20 incorrect combinations, the box will beep on 5 of them by sheer chance (on average). If you punch in all 131,115,985 possible combinations, then while the box is certain to beep for the one winning combination, it will also beep for 32,778,996 losing combinations (on average).
So this box doesn’t let you win the lottery, but it’s better than nothing. If you used the box, your odds of winning would go from 1 in 131,115,985 to 1 in 32,778,997. You’ve made some progress toward finding your target, the truth, within the huge space of possibilities.
Suppose you can use another black box to test combinations twice, independently. Both boxes are certain to beep for the winning ticket. But the chance of a box beeping for a losing combination is 1/4 independently for each box; hence the chance of both boxes beeping for a losing combination is 1/16. We can say that the cumulative evidence, of two independent tests, has a likelihood ratio of 16:1. The number of losing lottery tickets that pass both tests will be (on average) 8,194,749.
Since there are 131,115,985 possible lottery tickets, you might guess that you need evidence whose strength is around 131,115,985 to 1—an event, or series of events, which is 131,115,985 times more likely to happen for a winning combination than a losing combination. Actually, this amount of evidence would only be enough to give you an even chance of winning the lottery. Why? Because if you apply a filter of that power to 131 million losing tickets, there will be, on average, one losing ticket that passes the filter. The winning ticket will also pass the filter. So you’ll be left with two tickets that passed the filter, only one of them a winner. Fifty percent odds of winning, if you can only buy one ticket.
A better way of viewing the problem: In the beginning, there is 1 winning ticket and 131,115,984 losing tickets, so your odds of winning are 1:131,115,984. If you use a single box, the odds of it beeping are 1 for a winning ticket and 0.25 for a losing ticket. So we multiply 1:131,115,984 by 1:0.25 and get 1:32,778,996. Adding another box of evidence multiplies the odds by 1:0.25 again, so now the odds are 1 winning ticket to 8,194,749 losing tickets.
It is convenient to measure evidence in bits—not like bits on a hard drive, but mathematician’s bits, which are conceptually different. Mathematician’s bits are the logarithms, base 1/2, of probabilities. For example, if there are four possible outcomes A, B, C, and D, whose probabilities are 50%, 25%, 12.5%, and 12.5%, and I tell you the outcome was “D,” then I have transmitted three bits of information to you, because I informed you of an outcome whose probability was 1/8.
It so happens that 131,115,984 is slightly less than 2 to the 27th power. So 14 boxes or 28 bits of evidence—an event 268,435,456:1 times more likely to happen if the ticket-hypothesis is true than if it is false—would shift the odds from 1:131,115,984 to 268,435,456:131,115,984, which reduces to 2:1. Odds of 2 to 1 mean two chances to win for each chance to lose, so the probability of winning with 28 bits of evidence is 2/3. Adding another box, another 2 bits of evidence, would take the odds to 8:1. Adding yet another two boxes would take the chance of winning to 128:1.
So if you want to license a strong belief that you will win the lottery—arbitrarily defined as less than a 1% probability of being wrong—34 bits of evidence about the winning combination should do the trick.
In general, the rules for weighing “how much evidence it takes” follow a similar pattern: The larger the space of possibilities in which the hypothesis lies, or the more unlikely the hypothesis seems a priori compared to its neighbors, or the more confident you wish to be, the more evidence you need.
You cannot defy the rules; you cannot form accurate beliefs based on inadequate evidence. Let’s say you’ve got 10 boxes lined up in a row, and you start punching combinations into the boxes. You cannot stop on the first combination that gets beeps from all 10 boxes, saying, “But the odds of that happening for a losing combination are a million to one! I’ll just ignore those ivory-tower Bayesian rules and stop here.” On average, 131 losing tickets will pass such a test for every winner. Considering the space of possibilities and the prior improbability, you jumped to a too-strong conclusion based on insufficient evidence. That’s not a pointless bureaucratic regulation; it’s math.
Of course, you can still believe based on inadequate evidence, if that is your whim; but you will not be able to believe accurately. It is like trying to drive your car without any fuel, because you don’t believe in the fuddy-duddy concept that it ought to take fuel to go places. Wouldn’t it be so much more fun, and so much less expensive, if we just decided to repeal the law that cars need fuel?
Well, you can try. You can even shut your eyes and pretend the car is moving. But really arriving at accurate beliefs requires evidence-fuel, and the further you want to go, the more fuel you need.
此前，我将证据定义为「通过因果链与你想了解的事物纠缠在一起的事件」，将纠缠定义为「对目标的不同可能状态会有不同表现」。那么，需要多少纠缠——多少理性证据——才足以支持一个信念？
让我们从一个简单到可以用数学处理的问题开始：要赢得彩票，你需要把自己与彩票纠缠到什么程度？假设有七十个球，不放回地抽取，需要匹配六个数字才能获奖。那么共有 131,115,985 种可能的获奖组合，因此随机选取的一张票中奖概率为 1/131,115,985（0.0000007%）。要赢得彩票，你需要的证据选择性足以在 131,115,984 个备选方案中明显倾向于某一个组合。
假设有一些测试可以在获奖号码和落选号码之间做出概率性区分。例如，你可以在一个小黑盒子里输入一个组合，如果是获奖组合，它一定会鸣叫；如果组合不对，它有 1/4（25%）的概率鸣叫。用贝叶斯术语说，我们称似然比为 4:1。这意味着当我们输入正确组合时，盒子鸣叫的可能性是输入错误组合时的 4 倍。
仍然有大量可能的组合。如果你输入 20 个错误组合，盒子平均会对其中 5 个鸣叫（仅凭运气）。如果你输入所有 131,115,985 种可能的组合，那么虽然盒子必然会对那一个获奖组合鸣叫，但它也会对 32,778,996 个落选组合鸣叫（平均而言）。
所以这个盒子不能让你赢得彩票，但比没有强。如果你使用了这个盒子，你的中奖概率会从 1/131,115,985 提升到 1/32,778,997。你在巨大的可能性空间中朝着目标——真相——迈出了一些进展。
假设你可以使用另一个黑盒子对组合进行*两次**独立测试。两个盒子对获奖票都必然鸣叫。但一个盒子对落选组合鸣叫的概率是 1/4，对每个盒子独立而言；因此两个盒子都对落选组合鸣叫的概率是 1/16。我们可以说，两次独立测试的累积*证据的似然比为 16:1。通过两次测试的落选彩票平均会有 8,194,749 张。
由于共有 131,115,985 种可能的彩票，你可能猜测你需要的证据强度约为 131,115,985:1——一个事件或一系列事件，对于获奖组合来说发生的可能性是落选组合的 131,115,985 倍。实际上，这么多的证据只够给你一个均等的中奖机会。为什么？因为如果你将这么强大的过滤器应用于 1.31 亿张落选票，平均会有一张落选票通过过滤器。获奖票也会通过过滤器。所以你最终会得到两张通过过滤器的票，其中只有一张是赢家。如果你只能买一张票，那中奖概率是 50%。
更好的思考方式是：一开始，有 1 张获奖票和 131,115,984 张落选票，所以你的中奖赔率是 1:131,115,984。如果你使用一个盒子，它鸣叫的赔率对获奖票是 1，对落选票是 0.25。所以我们将 1:131,115,984 乘以 1:0.25，得到 1:32,778,996。再加一个证据盒子，赔率再乘以 1:0.25，现在赔率是 1 张获奖票对 8,194,749 张落选票。
用比特来衡量证据很方便——不是硬盘上的比特，而是数学家的比特，它们在概念上是不同的。数学家的比特是概率以 1/2 为底的对数。例如，如果有四种可能的结果 A、B、C 和 D，其概率分别为 50%、25%、12.5% 和 12.5%，而我告诉你结果是「D」，那么我向你传输了三比特的信息，因为我告知了你一个概率为 1/8 的结果。
恰好 131,115,984 略小于 2 的 27 次方。所以 14 个盒子或 28 比特的证据——一个对于票的假设为真时发生概率是为假时的 268,435,456:1 倍的事件——会将赔率从 1:131,115,984 移动到 268,435,456:131,115,984，化简为 2:1。赔率 2:1 意味着每一次输的机会对应两次赢的机会，所以凭 28 比特证据的中奖概率是 2/3。再加一个盒子，再加 2 比特证据，赔率就变成 8:1。再加两个盒子，中奖概率就达到 128:1。
所以如果你想要给自己颁发一个强烈相信自己会赢得彩票的许可——任意定义为出错概率低于 1%——关于获奖组合的 34 比特证据应该就够了。
总体而言，衡量「需要多少证据」的规则遵循类似的模式：假设所在的可能性空间越大，或者与其邻近假设相比假设看起来越不可能，或者你希望越有把握，你就需要越多的证据。
你无法违抗这些规则；你无法基于不充分的证据形成准确的信念。假设你有 10 个盒子排成一排，你开始向盒子输入组合。你不能在第一个获得所有 10 个盒子都鸣叫的组合处停下来，说：「但对于落选组合来说，发生这种事的概率是百万分之一！我就忽视那些象牙塔式的贝叶斯规则，停在这里。」平均而言，每找到一个赢家，就会有 131 张落选票通过这样的测试。考虑到可能性空间和先验的不可能性，你在不充分的证据基础上跳到了过于强烈的结论。这不是无意义的官僚规定；这是数学。
当然，如果这是你的心血来潮，你仍然可以基于不充分的证据相信；但你将无法准确地相信。这就像试图在没有燃料的情况下开车，因为你不相信汽车出行需要燃料这个迂腐的概念。如果我们决定废除汽车需要燃料的法律，那不是更有趣、更省钱吗？
好吧，你可以试试。你甚至可以闭上眼睛假装车在行驶。但真正抵达准确的信念需要证据-燃料，你想走得越远，你需要的燃料就越多。