为什么基线率在医学筛查里如此重要？
Why is the base rate so important in medical screening?
即使检测准确率95%，若疾病本身极罕见（基线率0.1%），阳性结果中真病人仍很少，这是贝叶斯更新。忽略基线率会严重高估患病概率。
Even with 95% accuracy, if a disease is rare (base rate 0.1%), few positives are true cases, by Bayes' update. Ignoring the base rate wildly overestimates your chance of being sick.
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
吴军,《数学之美》第3版,人民邮电出版社,2020
Wu Jun, Beauty of Mathematics, 3rd ed., Posts & Telecom Press, 2020
独立核验依据 · Independent verification: 核验依据 2 ISBN 9787115537973（豆瓣书目条目: 数学之美）
#基线率 #贝叶斯
冯·诺依曼-摩根斯特恩（VNM）效用定理怎样把“理性选择”变成“最大化期望效用”？
How does the von Neumann–Morgenstern (VNM) utility theorem turn 'rational choice' into 'maximize expected utility'?
VNM 定理在“完备性、传递性、连续性、独立性”四条理性公理下证明：一个人的风险偏好可表示为一个效用函数 u，使其在不确定选项间做选择等价于最大化期望效用 E[u]。独立性公理（复合赌局等价拆解）是关键，它把主观偏好变成可运算的数值，奠定现代决策论与博弈论。
Under four rationality axioms—completeness, transitivity, continuity, independence—the VNM theorem shows a person's risk preferences can be represented by a utility function u, and choosing among uncertain options is equivalent to maximizing expected utility E[u]. The independence axiom (compound lotteries split equivalently) is key, turning subjective preference into computable numbers and founding modern decision and game theory.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
von Neumann, J. 与 Morgenstern, O., 《博弈论与经济行为》(第2版), Princeton University Press
von Neumann, J. & Morgenstern, O., 'Theory of Games and Economic Behavior' (2nd ed.), Princeton University Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL1204974W：Theory of Games and Eco
#VNM效用定理 #期望效用 #理性公理 #博弈论
阿罗不可能定理说了什么？为什么投票也会无解？
What does Arrow's impossibility theorem say about voting?
阿罗证明：当选项不少于3且选民不少于2时，不存在一种既公平（尊重个人偏好、无关选项独立、非独裁）又能始终排出社会偏好的规则。说明集体决策有内在矛盾。
Arrow proved that with 3+ options and 2+ voters, no rule can be fair (respecting preferences, independent of irrelevant alternatives, non-dictatorial) and always yield a social ranking. Collective choice has inherent contradictions.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
阿罗《社会选择》
Arrow's theorem
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787563808403（豆瓣书目条目: 社会选择）
#阿罗定理 #投票
为什么第二价格密封拍卖（Vickrey 拍卖）会让竞拍者愿意报出自己的真实估值？
Why does a second-price sealed-bid auction (Vickrey auction) make bidders want to reveal their true valuation?
第二价格拍卖中，最高出价者中标但只付第二高的出价。此时无论别人报多少，你抬高自己的报价只会让你在“本不该赢”时风险变大、或“本该赢”时并不多得，所以最优策略就是按真实估值报价。维克里因此获 1996 年诺贝尔经济学奖，它也是在线广告竞价（如 Google AdWords）的基石。
In a second-price auction the highest bidder wins but pays the second-highest bid. Then, whatever others bid, raising your own bid only risks winning when you shouldn't (or gains nothing when you should), so the dominant strategy is to bid your true value. Vickrey won the 1996 Nobel Prize in Economics for this, and it underpins online ad auctions like Google AdWords.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Vickrey, W., 《投机、拍卖与竞争性密封投标》, Journal of Finance 16(1), 1961
Vickrey, W., 'Counterspeculation, Auctions, and Competitive Sealed Tenders', Journal of Finance 16(1), 1961
独立核验依据 · Independent verification: 核验依据 1 CROSSREF 标识 10.1111/j.1540-6261.1961.tb02789.x：COUNTERSP
#维克里拍卖 #第二价格 #优势策略 #真实报价
为什么期望收益为正，你也可能不该全押？
Why might you avoid betting even when expected profit is positive?
若赌注占你全部资产很大比例，一次失败就破产，这叫破产风险。凯利公式告诉我们应按比例下注，而非只要期望为正就全押。
If the stake is a large fraction of your wealth, one loss ruins you, called ruin risk. Kelly's formula says bet a proportion, not everything, just because expectation is positive.
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.
#破产风险 #凯利
为什么保险值得买，即使你大概率不理赔？
Why is insurance worth buying even if you likely won't claim?
因为保险买的是避免破产风险，而不是平均划算。小额保费换走大额灾难损失，对风险厌恶者是正向效用，体现的是期望效用而非期望金额。
Insurance buys protection from ruin, not average savings. A small premium offloads a huge loss, positive utility for the risk-averse, reflecting expected utility rather than expected money.
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.
#保险 #风险
选举里绝对多数和相对多数有什么不同？
What is the difference between an absolute and a relative majority?
相对多数（ plurality）是得票最多即可，哪怕未过半；绝对多数要求超过一半，否则可能进入第二轮。两者会影响少数派是否被代表。
A relative majority (plurality) wins with the most votes even under 50%; an absolute majority requires over half, often triggering a runoff. The rule affects how minorities are represented.
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 #多数规则
为什么平均往返速度不是两个速度的平均？
Why isn't average round-trip speed the average of two speeds?
同样路程，慢的那段占用更多时间，所以要用调和平均2ab/(a+b)才正确。平均速度看的是总路程除以总时间，而非算术平均。
The slower leg takes more time over the same distance, so the harmonic mean 2ab/(a+b) is correct. Average speed is total distance over total time, not the arithmetic mean.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#平均速度 #调和平均
买房或求职时，‘再等等会不会更好’如何用最优停止思考？
How can optimal stopping help with 'should I wait for something better' in house-buying or job-hunting?
你可以把前一段经历当作样本建立标准，之后遇到达标选项就行动，而不是无限观望。代价是可能错过更优者，但能避免永远等待的机会成本。
You can treat early options as a sample to set a standard, then act once a meeting option appears, rather than waiting forever. The cost is possibly missing a better one, but it avoids infinite opportunity cost.
难度 进阶 · 适合 大众
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
#最优停止 #生活
什么是期望值？为什么理性决策要看它？
What is expected value and why should rational decisions use it?
期望值是每种结果乘以其概率后加总，代表长期的平均收益。理性选择通常挑期望效用最高者，但人对风险的态度会修正它。
Expected value weights each outcome by its probability and sums them, the long-run average gain. Rational choice usually maximizes expected utility, though attitudes toward risk adjust it.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《概率论与数理统计》
Probability Theory and Mathematical Statistics (《概率论与数理统计》)
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787312018381（豆瓣书目条目: 概率论与数理统计）
#期望值 #决策
囚徒困境里，为什么两个“都为自已好”的选择加起来却是双输？
In the prisoner's dilemma, why do two 'self-interested' choices add up to a worse outcome for both?
囚徒困境中，每人单独看“背叛”都比“合作”划算（无论对方怎么选，背叛都不更差），于是理性人都选背叛，结果双方都获重刑——比都合作时的轻刑更糟。它说明个体理性未必导致集体最优，是博弈论与机制设计的起点。
In the prisoner's dilemma, 'defect' dominates 'cooperate' for each player individually (defect is never worse regardless of the other), so both rationally defect and both get a heavy sentence—worse than mutual cooperation. It shows individual rationality need not yield collective optimality, founding game theory and mechanism design.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Rapoport, A. 与 Chammah, A. M., 《囚徒困境》, University of Michigan Press
Rapoport, A. & Chammah, A. M., 'Prisoner's Dilemma', University of Michigan Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL23087512W：The University of Mich
#囚徒困境 #优势策略 #纳什均衡 #集体理性
为什么期望值相同时，人们的选择却不同？
Why do people choose differently when expected values are equal?
因为人看重确定性和损失厌恶：损失带来的痛苦大于同等收益的快乐，所以同样期望下多选择稳妥方案，这催生了期望效用理论。
People value certainty and are loss-averse: a loss hurts more than an equal gain pleases, so with equal expectations they prefer the safe option, giving rise to expected utility theory.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Tom、Fox、Trepel 与 Poldrack,《The Neural Basis of Loss Aversion in Decision-Making Under Risk》,Science,2007 (DOI: 10.1126/science.1134239)
Tom, Fox, Trepel & Poldrack, 'The Neural Basis of Loss Aversion in Decision-Making Under Risk', Science, 2007, DOI: 10.1126/science.1134239
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1126/science.1134239
#期望效用 #风险
为什么群体智慧有时准有时荒唐？
Why is the wisdom of crowds sometimes wise, sometimes foolish?
当个体判断独立且误差各异时，平均能抵消误差（准）；若大家互相模仿或受权威引导，就会集体犯错（荒唐）。关键在于独立性。
When individual judgments are independent with varied errors, averaging cancels them out (wise); if people imitate each other or follow authority, the crowd errs together (foolish). Independence is the key.
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.
#群体智慧 #独立
三位选民对候选人A、B、C的偏好分别是A>B>C、B>C>A、C>A>B，为什么此时两两多数决会得出A胜B、B胜C、C又胜A的循环？这说明简单多数原则存在什么根本缺陷？
Three voters rank candidates A, B, C as A>B>C, B>C>A, and C>A>B respectively; why does pairwise majority voting then produce the cycle A beats B beats C beats A, and what fundamental flaw of simple majority rule does this reveal?
每位候选人在一对一比较中都能拿到过半票：A以2比1胜B，B以2比1胜C，C又以2比1胜A，于是社会偏好出现循环，不存在孔多塞赢家。即使所有个体排序都完全传递，集体决策却可能不传递，掌握议程的人由此得以操纵结果。这一悖论直接推动了阿罗不可能定理与社会选择理论的系统研究。
Each candidate wins some pairwise contest by two votes to one: A beats B, B beats C, and C beats A, so the social preference forms a cycle and no Condorcet winner exists. Even when every individual ranking is perfectly transitive, the collective outcome may fail to be transitive, letting whoever controls the agenda steer the result. This paradox directly motivated Arrow's impossibility theorem and the systematic study of social choice.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
孔多塞《应用概率分析考察多数表决决策之试论》（1785）；肯尼斯·阿罗《社会选择与个人价值》（1951）
Marquis de Condorcet, 'Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix' (1785); Kenneth Arrow, 'Social Choice and Individual Values', Cowles Commission monograph (1951)
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL5103552W：Social choice and indiv
#投票悖论 #社会选择
为什么价格降50%再涨50%回不到原点？
Why doesn't a 50% drop then 50% rise return to the start?
100元降50%剩50元，再涨50%只是75元，因为基数变了。涨跌百分比不对称，这是复利与相对变化的基本陷阱。
100 down 50% leaves 50; up 50% then is only 75, because the base changed. Percentage gains and losses are asymmetric, a basic trap in compounding and relative change.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Beckmann, S., 《中小学数学教师数学》(第4版), Pearson
Beckmann, S., 'Mathematics for Elementary Teachers' (4th ed.), Pearson
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL18708410W：Mathematics for elemen
#百分比 #复利
为什么沉没成本不该影响决策？
Why shouldn't sunk costs affect your decisions?
已经发生、无法收回的钱（如看了一半的烂片票）不该左右未来选择。理性决策只看边际成本与收益，也就是往前看。
Money already spent and unrecoverable (like a half-watched bad movie ticket) should not sway future choices. Rational decisions look only at marginal costs and benefits, i.e. forward.
难度 入门 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Arkes 与 Blumer,《The psychology of sunk cost》,Organizational Behavior and Human Decision Processes,1985 (DOI: 10.1016/0749-5978(85)90049-4)
Arkes & Blumer, 'The psychology of sunk cost', Organizational Behavior and Human Decision Processes, 1985, DOI: 10.1016/0749-5978(85)90049-4
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1016/0749-5978(85)90049-4
#沉没成本 #决策
促销说第二件半价，和直接打七五折一样吗？
Is 'second item half price' the same as a 25% discount?
两者数学上都是 75 折，也就是便宜 25%，但第二件半价用锚定与赠予感制造划算错觉，说明营销利用的是心理而非纯算术。
Mathematically both come to 75% of the price, that is a 25% discount, but 'second half price' uses anchoring and a gift feeling to seem like a better deal, showing marketing exploits psychology, not just arithmetic.
难度 入门 · 适合 幼儿(3-6)
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.
#促销 #心理