两个信封一个装x元另一个装2x元，你打开看到y元后，换信封的期望是1.25y元，那是否该一直换？
Two envelopes contain x and 2x; after seeing y you compute the other's expected value as 1.25y. Should you always switch?
不能，因为推导隐含假设看到的y可对应任意x的两种情形，但先验分布未定义导致期望计算无效。它揭示了无界分布下期望可能失真。
No; the calculation silently assumes y could be the smaller or larger amount for arbitrary x, but an undefined prior makes the expectation meaningless. It shows expectations can mislead under unbounded distributions.
难度 专业 · 适合 高中(15-18)
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
#双信封 #期望 #悖论
什么是伯克森悖论，为什么医院里两种病看起来负相关其实未必？
What is Berkson's paradox, and why might two diseases appear negatively correlated in a hospital when they are not?
医院样本以至少患一种病为入院条件，会造成入院人群里两种病的共存被系统性低估，呈现虚假负相关。这是选择偏差的典型例子。
Hospital samples condition on having at least one disease, which systematically undercounts co-occurrence and creates a spurious negative correlation — a classic selection bias.
难度 专业 · 适合 高中(15-18)
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
#伯克森 #选择偏差 #统计
圣彼得堡游戏的期望收益为何会发散，它揭示了什么？
Why does the St. Petersburg game have infinite expected value, and what does it reveal?
游戏奖金随抛到首次正面的次数指数增长，期望为无穷大，但没人愿付高价参与。它推动了期望效用理论的发展，说明期望金额不等于主观价值。
The payout grows exponentially with the waiting time, so expected value is infinite, yet nobody pays a fortune to play. It motivated expected utility theory: expected money is not the same as subjective value.
难度 专业 · 适合 高中(15-18)
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
#圣彼得堡 #期望 #效用
在圆内随机取一条弦，其长度超过内接等边三角形边长的概率是多少？为什么答案不唯一？
For a chord chosen at random in a circle, what is the probability it is longer than a side of the inscribed equilateral triangle, and why is the answer not unique?
答案取决于随机的定义，可以是1/3、1/2或1/4。它说明在连续概率中必须先明确概率测度，否则随机是模糊的。
The answer depends on how random is defined: 1/3, 1/2, or 1/4. It shows that in continuous probability you must first fix a probability measure.
难度 专业 · 适合 高中(15-18)
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
#伯特兰悖论 #几何概率 #测度
什么是检察官谬误，为什么匹配概率极小不等于被告无罪概率极小？
What is the prosecutor's fallacy, and why does a tiny match probability not mean a tiny probability the defendant is innocent?
它把在随机人中特征匹配的概率误当成被告无罪时匹配的概率。要结合基础人数用贝叶斯定理，才能正确评估被告有罪的概率。
It confuses the probability of a match given innocence with the probability of innocence given a match. Bayes' theorem with the base rate is needed for the correct guilt probability.
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
#检察官谬误 #贝叶斯 #法庭统计
什么是辛普森悖论，为什么分组看和合并看结论会相反？
What is Simpson's paradox, and why can grouped and combined data lead to opposite conclusions?
当数据按某个变量分层后，各层的趋势与汇总后的整体趋势可能完全相反。它常因各层样本量悬殊、混杂变量未被控制而产生。
In Simpson's paradox, trends in subgroups can reverse in the pooled data, usually because of unequal group sizes and an uncontrolled confounder.
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
#辛普森悖论 #统计 #混淆变量
已知某人有两孩且其中至少一个是男孩，两个都是男孩的概率是多少？若再加上星期二出生的条件呢？
A family has two children and at least one is a boy. What is the probability both are boys? And what if we also learn that boy was born on a Tuesday?
仅知至少一男孩时是1/3；加上星期二男孩这一更具体的信息后，概率变为13/27。额外细节改变了样本空间，从而改变了结果。
With only at least one boy the chance is 1/3; adding born on a Tuesday changes it to 13/27. The extra detail reshapes the sample space.
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
#星期二男孩 #条件概率 #贝叶斯
某病患病率1%，检测准确率99%，若检测呈阳性，你真患病的概率是多少？
A disease affects 1% of people and a test is 99% accurate. If you test positive, what is the chance you actually have the disease?
约50%。大量健康人被误报，阳性中真病人被稀释，必须用贝叶斯结合基础患病率。
About 50%. Most people are healthy, and even a 1% false-positive rate produces many false alarms, so Bayes' theorem with the base rate is essential.
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
#贝叶斯 #假阳性 #基础率
三囚犯中已知将有一人获释，你问守卫B会被处决吗，得知是后你获释的概率变了吗？
Of three prisoners one will be freed; you ask the guard will B be executed and hear yes. Did your chance of freedom change?
你（A）的获释概率从1/3升到1/2，而C的概率也升到1/2。这与蒙提霍尔相似，守卫的回答提供了额外信息。
Your (A's) chance of freedom rises from 1/3 to 1/2, as does C's. Like Monty Hall, the guard's answer supplies new information.
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
#三囚犯 #条件概率 #信息
蒙提霍尔（三门）问题该不该换门？
Should you switch doors in the Monty Hall (three-door) problem?
应该换。你最初选中的概率是1/3，奖品在另两扇门后的概率是2/3；主持人揭示一扇空门后，这2/3全转移到剩下那扇门上，换门胜率翻倍。
Yes, switch. Your first pick has 1/3 chance; the prize is behind the other two doors with 2/3. After the host reveals an empty door, that 2/3 transfers to the remaining door, doubling your win chance.
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
#随机 #反直觉
大数定律说抛硬币正面比例会趋近1/2，这是否意味着少量抛掷也会接近？
The law of large numbers says the heads proportion approaches 1/2, but does it mean a small number of tosses will also be close?
不会，大数定律只保证长期平均收敛，短期波动仍可能很大。样本越多，比例才越稳定地贴近真值。
No; it only guarantees long-run convergence of the average. Short-term fluctuations can still be large, and stability improves only with more trials.
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
#大数定律 #频率 #概率
抛硬币连续出现5次正面后，下一次更可能是反面吗？
After five heads in a row, is the next fair-coin toss more likely to be tails?
不会，每次抛掷独立，正面概率仍是1/2。赌徒谬误是把长期会平衡错误地当成短期会补偿。
No; each toss is independent and still has probability 1/2 for heads. The gambler's fallacy wrongly treats long-run balance as short-run compensation.
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 #独立性 #概率
生日悖论在说什么？
What does the birthday paradox say?
在23人中，两人生日相同的概率已超过50%。因为要比较的是‘两两组合’约253对，而非与某特定人比较，所以冲突来得比直觉快得多。
In a group of 23, the chance two share a birthday exceeds 50%. We compare ~253 pairs, not against one fixed person, so collisions arrive far faster than intuition suggests.
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
#随机 #概率