辛普森悖论为什么能让结论完全反转？
Why can Simpson's paradox flip a conclusion completely?
当数据分组时A优于B，但合并后B反而优于A，往往因为各组样本量悬殊。它提醒我们：只看汇总数字或只看分组，都可能被骗。
Data can show A better than B within groups yet B better overall once combined, often because group sizes differ sharply. It warns that both pooled and grouped views can mislead.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Pearl、Glymour 与 Jewell《Causal Inference in Statistics: A Primer》（Wiley）
Pearl, Glymour and Jewell, Causal Inference in Statistics: A Primer, Wiley
独立核验依据 · Independent verification: 核验依据 1 ISBN 978-1-119-18687-8 (2016, Wiley)
#辛普森悖论 #统计
意想不到的绞刑悖论说的是什么？
What is the unexpected hanging paradox?
法官说周末前某天绞刑且当天早上你不知情，囚犯推理出不可能发生，结果却在出乎意料的那天执行。它揭示了自指预言与知识逻辑的微妙矛盾。
A judge says the hanging will occur on a weekday and be a surprise; the prisoner deduces it cannot happen, yet it does. It exposes subtle conflicts in self-referential prediction and knowledge.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Chow,《The Surprise Examination or Unexpected Hanging Paradox》,The American Mathematical Monthly,1998 (DOI: 10.1080/00029890.1998.12004847)
Chow, 'The Surprise Examination or Unexpected Hanging Paradox', The American Mathematical Monthly, 1998, DOI: 10.1080/00029890.1998.12004847
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1080/00029890.1998.12004847
#悖论 #自指
双信封悖论为什么让人左右为难？
Why does the two-envelope paradox leave you stuck?
一个信封的钱是另一个的两倍，你算期望后总想换，但换完用同样逻辑又想换回去，陷入循环。它暴露了期望值在信息不足、对称设定下的自相矛盾。
One envelope holds twice the other; computing expected value makes you want to switch, but the same logic then makes you want to switch back, an infinite loop. It exposes how expected value can contradict itself under symmetric incomplete information.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#双信封 #期望值
巴拿赫-塔斯基悖论说能把球分成可重组的两个球，怎么可能？
Banach-Tarski says a ball can be split and reassembled into two balls - how?
它利用选择公理把球切成极诡异的不可测点集，再仅靠旋转和平移拼成两个同样大的球。这不是物理上可操作的，而是说明体积对这类点集会失效，揭示无穷与测量的边界。
It uses the axiom of choice to cut a ball into bizarre non-measurable point sets that, by rotation and translation only, reassemble into two equal balls. Not physically doable, it shows volume fails for such sets, revealing limits of infinity and measure.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#巴拿赫塔斯基 #悖论
“秘书问题”（最优停止）为什么说先看放弃前约 37% 的候选再选，成功率最高？
In the 'secretary problem' (optimal stopping), why does rejecting the first ~37% of candidates then picking the next best maximize success?
在只能顺序面试、当场决定且不能回头的设定下，最优策略是跳过前 r−1 个（用于建立标准），之后一旦出现比前面都好的就录用。数学上最优分界约为 n/e ≈ 37%，此时选中“全局最佳”的概率约 1/e ≈ 37%。它把“何时停止”转化为漂亮的 1/e 常数，是效用与信息权衡的经典范例。
With sequential interviews, irrevocable choices, and no recall, the optimal rule skips the first r−1 to set a standard, then takes the first candidate better than all before. The optimal cutoff is about n/e ≈ 37%, giving a ~1/e ≈ 37% chance of the overall best—turning 'when to stop' into the elegant constant 1/e, a classic example of the utility-information trade-off.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Ferguson, T. S., 《谁解开了秘书问题？》, Statistical Science 4(3), 1989
Ferguson, T. S., 'Who Solved the Secretary Problem?', Statistical Science 4(3), 1989
独立核验依据 · Independent verification: 核验依据 1 CROSSREF 标识 10.1214/ss/1177012493：Who Solved the Secreta
#秘书问题 #最优停止 #1/e #37%法则
为什么样本越大越准，却也可能更暴露极端个案？
Why do larger samples get more accurate yet expose more extremes?
样本越大，罕见事件越可能出现（如百万分之一的副作用在千万人中会出现几十例），所以大数据既更可靠又更容易显出极端个案，需要正确解读。
Larger samples make rare events more likely to appear (a one-in-a-million effect shows up dozens of times in tens of millions), so big data is both more reliable and more likely to surface extremes, needing careful interpretation.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#大数据 #小概率
为什么23个人里，两人生日相同的概率超过50%？
Why is the chance two people share a birthday over 50% with just 23 people?
比较的是任意两人的配对，而不是某人和你。23人共有253对，配对数量增长极快，所以概率很快超过一半。这就是生日悖论：直觉低估了配对数。
It compares any pair among the group, not you versus others. With 23 people there are 253 pairs, so the probability of a match climbs past half quickly. The birthday paradox shows intuition underestimates pairings.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#生日悖论 #概率
为什么投票给小党不一定是浪费？
Why isn't voting for a small party necessarily wasted?
在比例代表制下，小党得票累积可获席位并影响政策；即便在单选区，弃保效应和关键少数也让小党有杠杆。这是投票数学与制度的互动。
Under proportional representation, a small party's cumulative votes win seats and shape policy; even in single-seat districts, coordination effects and swing minorities give small parties leverage. It is the interplay of voting math and institutions.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#投票 #小党
轮盘赌里“每次输了就加倍下注”的马丁格尔策略，为什么长期一定亏？
In roulette, why does the martingale strategy ('double after every loss') inevitably lose in the long run?
马丁格尔法靠“输后加倍”保证一回本就净赚一手本金，但每次都需要比上一次多一倍的资金；一旦遇到长连输，所需本金会超过桌限或你的全部家当，于是在有限财富下迟早被迫停手，此前累积的小赢一次清零。数学期望上它仍为零和偏负（抽水），只是把亏损推迟成大崩盘。
The martingale recovers one unit by doubling after each loss, but each step needs double the capital; a long losing streak exhausts the table limit or your wealth, forcing you to stop and wipe out prior small gains. Its expected value stays zero or negative (house edge), merely delaying loss into a rare crash.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Dubins, L. E. 与 Savage, L. J., 《必败的赌博：随机过程不等式》, McGraw-Hill
Dubins, L. E. & Savage, L. J., 'How to Gamble If You Must: Inequalities for Stochastic Processes', McGraw-Hill
独立核验依据 · Independent verification: 核验依据 1 ARCHIVE 标识 howtogambleifyou0000dubi：How to gamble if you
#马丁格尔 #轮盘赌 #加倍策略 #破产
拍卖里的“赢家诅咒”为什么让出价最高的人反而最可能吃亏？
Why does the 'winner's curse' in auctions mean the highest bidder is most likely to lose out?
在价值不确定的共同价值拍卖中，每个竞拍者只凭自己的估计出价；出价最高的人，往往正是把价值估得最高（最可能高估）的那位，中标后才发现真实价值低于自己付的价。因此“赢”了拍卖却常“亏”了钱，理性买家应故意压低出价。
In a common-value auction with uncertain worth, each bids on their own estimate; the winner is typically the one who overestimated most, discovering post-win that the true value is below the price paid. So winning the auction often means losing money, and rational bidders shade their bids down.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
Capen, E. C., Clapp, R. V. 与 Campbell, W. M., 《高风险情形下的竞争性投标》, AAPG Bulletin 56(4), 1971
Capen, E. C., Clapp, R. V. & Campbell, W. M., 'Competitive Bidding in High-Risk Situations', AAPG Bulletin 56(4), 1971
#赢家诅咒 #共同价值拍卖 #过度自信 #反直觉
为什么0.999…看起来小于1，其实等于1？
Why does 0.999… equal 1 even though it looks smaller?
设x=0.999…，则10x=9.999…，两式相减得9x=9，所以x=1。或者说1/3=0.333…，乘3得到1=0.999…。无限小数没有最后的9，二者是同一实数的两种写法。
Let x=0.999…; then 10x=9.999…, subtract to get 9x=9, so x=1. Or 1/3=0.333… and times 3 gives 1=0.999…. There is no final 9, so they are two names for one real number.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#趣味数学 #无限小数
蒙提霍尔问题：换门中奖率真的更高吗？
Monty Hall problem: is switching doors really better?
是的。你初选正确的概率是1/3；主持人揭开一只空门后，剩下那扇门正确的概率变成2/3，所以换门把胜率从1/3提高到2/3。
Yes. Your first pick is right only 1/3 of the time; after the host reveals an empty door, the remaining door is right 2/3 of the time, so switching doubles your win rate from 1/3 to 2/3.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#蒙提霍尔 #概率
为什么真正随机其实很难？
Why is true randomness actually hard?
人编出的数字序列常有规律（避免过长连号），算法随机数多为伪随机（决定性公式），真正随机需要物理噪声。这也是密码学关心的话题。
Human-made sequences show patterns (avoiding long runs) and algorithmic randoms are usually pseudorandom (deterministic formulas); true randomness needs physical noise. That is why cryptography cares.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#伪随机 #随机
为什么过去业绩好的基金不代表未来好？
Why don't past top-performing funds predict future success?
在随机波动中总有一部分基金某年表现佳，若只宣传幸存者会误导。长期看多数主动基金跑不赢指数，这是统计与选择偏差的结果。
Random fluctuation always produces some funds that shine in a given year; advertising only survivors misleads. Long run, most active funds underperform the index, a result of statistics and selection bias.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#基金 #统计
为什么无限旅馆永远不会客满？
Why can Hilbert's infinite hotel never be full?
希尔伯特设想一个有可数无限个房间且全满的旅馆；新来一位客人时，让1号去2号、2号去3号……腾出1号。这说明无限集合很反直觉：一部分能和集合本身一样大。
Hilbert imagined a fully occupied hotel with countably infinite rooms; a new guest is housed by moving room n to n+1, freeing room 1. It shows infinite sets are counterintuitive: a part can be as large as the whole.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#无限旅馆 #无穷
为什么连续抛10次正面几乎不可能，但某段10连却必然出现？
Why is 10 heads in a row near impossible yet some 10-streak inevitable?
指定某10次序列的概率是(1/2)^10约千分之一，极罕见；但抛很多次后，几乎必然会出现某一段10连。这是指定事件与存在事件的区别。
A specified 10-toss sequence has probability about 1/1024, very rare; but after many tosses some 10-streak is almost certain to appear. That is the difference between a specified and an existing event.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
W. Feller《概率论导论及其应用》第1卷, Wiley, 1950
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1950
独立核验依据 · Independent verification: 核验依据 1 ISBN 9789994311071 (1968) · 核验依据 2 ISBN 9780821807491 (1997) · 核验依据 3 ISBN 9789994311071 (1991)
#随机序列 #概率
连续抛硬币出现五次正面后，下一次正面的概率还是1/2吗？
After five heads in a row, is the next coin flip still 1/2 heads?
是的。硬币没有记忆，每次抛掷相互独立，之前的结果不影响下一次。觉得该出反面了，是赌徒谬误。
Yes. A coin has no memory; each toss is independent, so past results do not affect the next. Thinking tails is 'due' is the gambler's fallacy.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#赌徒谬误 #独立
什么是幸存者偏差？它怎么让人误判？
What is survivorship bias and how does it mislead us?
我们只看到活下来的样本（如返航战机上没中弹的部位），忽略了被淘汰的（被击落的），于是得出错误结论。二战统计学家瓦尔德据此建议加固无弹孔处。
We only see survivors (e.g. returning planes' intact spots) and ignore the eliminated (shot-down planes), leading to wrong conclusions. Statistician Wald thus advised reinforcing the unhit areas.
难度 入门 · 适合 小学高段(9-12)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#幸存者偏差 #统计
抛两枚硬币，正正和正反哪个更可能？
Tossing two coins, which is more likely: two heads or one of each?
抛两枚有正正、正反、反正、反反四种等可能结果。一正一反包含正反和反正两种，所以概率1/2，比正正的1/4更可能；顺序会误导直觉。
Two coins give HH, HT, TH, TT, all equally likely. One head one tail covers HT and TH, so its chance is 1/2, more than HH's 1/4; order tricks our intuition.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#硬币 #概率
为什么平均数会骗人？平均数和中位数差很多说明什么？
Why can the average lie, and what does a big gap with the median mean?
少数极大的值会拉高平均数（如贫富差距），使它远高于多数人的真实情况。看收入、房价时，中位数往往更能代表普通人。
A few huge values pull the average up (e.g. wealth gaps), making it far above most people's reality. For incomes or house prices the median usually represents the typical person better.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#平均数 #统计陷阱
为什么买彩票差点中奖其实并不稀奇？
Why is almost winning the lottery not actually rare?
中奖组合只有一种，但差一个数字的组合有成千上万种，所以几乎每次开奖都有大量差一点的票；这并不说明你运气好，只是零事件附近的近邻很多。
There is only one winning combination, but millions of combinations are off by a single number, so nearly every draw produces many near-misses. That does not mean you were lucky; near-misses are just numerous.
难度 入门 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#彩票 #概率错觉