谷歌的 PageRank 和马尔可夫链有什么关系？
How is Google's PageRank related to Markov chains?
PageRank 把网络浏览者抽象成一个马尔可夫链：状态是网页，转移按网页上的链接等概率进行。PageRank 值正是该链的平稳分布——长期看访问每个页面的比例。为避免“死胡同”和循环陷阱，拉里·佩奇引入阻尼因子（约 0.85），让浏览者偶尔随机跳到任意网页，从而保证平稳分布存在且唯一，再按它给网页排序。
PageRank models a web surfer as a Markov chain: states are pages and transitions follow the page's links with equal probability. The PageRank value is the chain's stationary distribution—the long-run proportion of visits to each page. To handle dead ends and trapping cycles, Larry Page introduced a damping factor (about 0.85) letting the surfer occasionally jump to a random page, guaranteeing a unique stationary distribution used to rank pages.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：此条未指向带版次的可引用著作（通识性内容、机构文件或未展开的英文书名），本库不提供独立书目出处；如需精确引用，需由编辑补全具体篇目与版次。
arXiv（PageRank 原论文）
arXiv (original PageRank paper)
#PageRank #马尔可夫链 #平稳分布
为什么一维、二维随机游走迟早回到原点，而三维却不一定？
Why do 1D and 2D random walks return to the origin almost surely, but not in 3D?
这是波利亚定理（1921）。在一维和二维，随机游走是“常返”的：回到原点的概率为 1，而且会无限多次返回。三维中回到原点的概率只有约 34%，因此是“暂态”的。直观原因是维度越高可选方向越多，回到原点所需的巧合被大量稀释。它为扩散、高分子构象和网络上的随机搜索提供了数学模型。
This is Pólya's theorem (1921). In one and two dimensions the walk is recurrent: the probability of returning to the origin is 1, and it returns infinitely often. In three dimensions the return probability is only about 0.34, so the walk is transient. Intuitively, higher dimension offers more directions, diluting the coincidence needed to come back. This provides mathematical models for diffusion, polymer conformations, and random search on networks.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Pólya,《Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz》,Mathematische Annalen,1921 (DOI: 10.1007/bf01458701)
Pólya, 'Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz', Mathematische Annalen, 1921, DOI: 10.1007/bf01458701
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1007/bf01458701
#随机游走 #常返性 #波利亚定理
大偏差原理（large deviation principle）如何量化“极小概率事件”的指数衰减速度？
How does the large deviation principle quantify the exponential decay rate of very small probabilities?
大偏差原理用“速率函数” I(x) 描述随机变量偏离均值的概率：P(Sₙ/n ≈ x) ≈ e^(−nI(x))。它比中心极限定理更精细地刻画了尾部，说明极端偏离的概率随样本量 n 呈指数衰减；在统计物理、风险与通信中有广泛应用，源自理查德·埃利斯等人对 Cramér 定理的推广。
The large deviation principle describes rare deviations via a rate function I(x): P(Sₙ/n ≈ x) ≈ e^(−nI(x)). It refines the central limit theorem by capturing the tail, showing extreme deviations decay exponentially in the sample size n, with wide use in statistical physics, risk, and communications—built on Cramér's theorem.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Dembo, A. 与 Zeitouni, O., 《大偏差技术及其在统计中的应用》, Springer
Dembo, A. & Zeitouni, O., 'Large Deviations Techniques and Applications', Springer
独立核验依据 · Independent verification: 核验依据 1 CROSSREF 标识 10.1007/978-1-4612-5320-4：Large Deviations T
#大偏差原理 #速率函数 #指数衰减 #Cramér定理
什么是鞅（martingale）？
What is a martingale?
鞅是一种随机过程，其未来条件的期望等于当前值（无“免费趋势”）。在公平赌博中财富就是鞅，这也说明“加倍下注”无法保证获利。
A martingale's conditional future expectation equals its present value; fair-game wealth is a martingale, so doubling bets can't guarantee profit.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
美国数学会(AMS)《随机过程导论》
American Mathematical Society (AMS), Introduction to Stochastic Processes
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787111315445（豆瓣书目条目: 随机过程导论）
#概率 #随机过程
随机游走与赌徒破产是什么？
What are random walks and gambler's ruin?
随机游走是每步随机前进或后退的路径；赌徒破产问题证明：在公平赌博中，资金有限的赌徒最终必然输光，无论起始资金多少。
A random walk moves randomly each step; gambler's ruin shows a finite-bankroll player eventually goes broke with probability 1 in fair play.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Grinstead, C. M. 与 Snell, J. L., 《概率导论》, 美国数学会 (AMS)
Grinstead, C. M. & Snell, J. L., 'Introduction to Probability', American Mathematical Society
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780821807491 (1997)
#概率 #随机过程
什么是大数定律？
What is the law of large numbers?
大数定律说当试验次数足够多时，事件发生的频率会稳定地接近其真实概率。例如抛很多次硬币，正面比例会趋近于1/2。
The law of large numbers says frequencies converge to true probabilities as the number of trials increases.
难度 专业 · 适合 高中(15-18)
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）
#概率 #大数定律
中心极限定理为什么被称为统计学的基石？
Why is the Central Limit Theorem called the cornerstone of statistics?
中心极限定理指出：大量相互独立、同分布（且方差有限）的随机变量之和，其标准化分布会趋近于正态分布，无论原始分布是什么形状。这解释了为何自然界与社会中大量「均值类」现象呈钟形曲线，并使基于正态分布的置信区间、假设检验等推断方法拥有了普遍适用的理论依据。
The Central Limit Theorem states that the standardized sum of many independent, identically distributed random variables (with finite variance) tends toward a normal distribution, regardless of the original shape. This explains why so many natural and social 'average-like' phenomena form bell curves, and gives the theoretical backbone for confidence intervals, hypothesis tests, and other normal-based inference.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#中心极限定理 #正态分布 #样本均值 #统计推断
双信封悖论是什么？
What is the two-envelope paradox?
两信封各装钱，一个正好是另一个的两倍。你拿一个后想“换”可能赚一倍也可能亏一半，期望却算成 1.25 倍——诱导你无限换。错误源于对任意金额都假设了对称分布。
Two envelopes, one with twice the other's amount; switching seems to gain 25% in expectation—but the same logic repeats forever. The flaw is assuming a symmetric distribution for any observed amount.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
Horgan,《The Two‐Envelope Paradox, Nonstandard Expected Utility, and the Intensionality of Probability》,Noûs,2000 (DOI: 10.1111/0029-4624.00279)
Horgan, 'The Two‐Envelope Paradox, Nonstandard Expected Utility, and the Intensionality of Probability', Noûs, 2000, DOI: 10.1111/0029-4624.00279
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1111/0029-4624.00279
#概率 #双信封
马尔可夫链刻画的是一类怎样的随机过程？
What kind of stochastic process do Markov chains describe?
马尔可夫链是一种随机过程，其下一个状态只依赖于当前状态，而与更早的历史无关——这称为「无记忆性」或马尔可夫性质。它由安德烈·马尔可夫在 20 世纪初提出，用于建模语言、股价和网页跳转等序列。PageRank 本质上就是在万维网上随机游走的马尔可夫链平稳分布。
A Markov chain is a stochastic process whose next state depends only on the current state, not on earlier history—the 'memoryless' or Markov property. Introduced by Andrey Markov in the early 20th century, it models sequences like language, stock prices, and web navigation. PageRank is essentially the stationary distribution of a random walk (Markov chain) over the web.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
美国数学学会（AMS）；Sheldon Ross《随机过程》
American Mathematical Society (AMS); Ross, Stochastic Processes
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL65235W：Stochastic processes（首版 1
#马尔可夫链 #无记忆性 #随机过程 #平稳分布
睡美人悖论是什么？
What is the Sleeping Beauty paradox?
睡美人被唤醒并问硬币结果；若正面她被唤醒一次，反面被唤醒两次（并遗忘）。问她此刻猜正面的概率？“Thirder”认为是 1/3，“halfer”认为是 1/2——这是关于如何定义概率的哲学分歧。
Sleeping Beauty is awakened and asked the coin outcome; she's woken once on heads, twice on tails (with amnesia). 'Thirders' say the chance of heads now is 1/3, 'halfers' say 1/2—a debate about how to define probability.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
CHEN,《Another Perspective of the Sleeping Beauty Problem: What Lessons Can We Learn from the Sleeping Beauty Problem?》,International Conference on Computers in Education,2019 (DOI: 10.58459/icce.2019.518)
CHEN, 'Another Perspective of the Sleeping Beauty Problem: What Lessons Can We Learn from the Sleeping Beauty Problem?', International Conference on Computers in Education, 2019, DOI: 10.58459/icce.2019.518
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.58459/icce.2019.518
#概率 #睡美人
什么是马尔可夫链？
What is a Markov chain?
马尔可夫链是一串随机变量，下一步的状态只取决于当前状态（无记忆性）。它用于排队、网页排名和天气预测。
A Markov chain's next state depends only on the current one (memoryless); used in queues, PageRank and weather models.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
美国数学会(AMS)《随机过程导论》
American Mathematical Society (AMS), Introduction to Stochastic Processes
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787111315445（豆瓣书目条目: 随机过程导论）
#概率 #随机过程
圣彼得堡悖论是什么？
What is the St. Petersburg paradox?
一枚公平硬币反复抛，直到第一次正面出现；若第 n 次才正面，你赢得 2ⁿ 元。期望收益是无穷大，但没人愿付高价玩——揭示了“期望效用”比“期望金额”更合理。
A game pays 2ⁿ if the first head appears on toss n; expected payoff is infinite, yet nobody pays much to play—showing expected utility beats expected value.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#概率 #圣彼得堡
为什么高维积分特别适合用蒙特卡洛方法？
Why is Monte Carlo especially suited to high-dimensional integrals?
网格法在 d 维空间中每个方向剖分 n 段，共需要 n^d 个点，随维数指数爆炸，这就是“维数灾难”。蒙特卡洛用随机抽样估计积分，由中心极限定理，误差与 √N 成反比，与维数无关。因此从统计物理、金融期权定价到贝叶斯推断，处理高维问题时蒙特卡洛几乎是唯一可行的路径。
Grid methods in d dimensions need n^d points when each direction is divided into n segments, exploding exponentially with dimension—the curse of dimensionality. Monte Carlo estimates integrals by random sampling; by the central limit theorem the error decays like 1/√N, independent of dimension. So for high-dimensional problems—statistical physics, option pricing, Bayesian inference—Monte Carlo is often the only feasible route.
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
#蒙特卡洛方法 #高维积分 #维数灾难
什么是贝叶斯定理？
What is Bayes' theorem?
贝叶斯定理由已知结果反推原因的概率：P(A|B) = P(B|A)·P(A) / P(B)，是统计推断与机器学习的理论核心。
Bayes' theorem updates the probability of a cause given an effect: P(A|B)=P(B|A)P(A)/P(B); core of inference and ML.
难度 专业 · 适合 高中(15-18)
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）
#概率 #贝叶斯
鞅的停时定理（可选抽样定理）在什么条件下成立？为什么“公平赌局下任何停止策略都无法获利”这一直觉需要这些条件？
Under what conditions does the optional stopping theorem for martingales hold, and why does the intuition that no stopping strategy beats a fair game require those conditions?
可选抽样定理通常要求停时有界、或停时的期望有限且鞅增量一致有界等条件之一；此时 E[M_T] = E[M_0]。缺少这些条件时结论可能失效：例如在公平赌局中采用“翻倍加注直到赢”的策略，停时期望虽为正，但对应停时有无穷期望资金需求，定理前提被破坏。
The optional stopping theorem holds under conditions such as a bounded stopping time, or an integrable stopping time together with uniformly bounded martingale increments; then E[M_T] = E[M_0]. Without them the conclusion can fail: in a fair game, the martingale betting system—doubling stakes until a win—has positive expected gain at stopping, but the required capital has infinite expectation, violating the theorem's hypotheses.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Williams, D. (1991). Probability with Martingales. Cambridge University Press, Chapter 10.
Williams, D. (1991). Probability with Martingales. Cambridge University Press, Chapter 10.
独立核验依据 · Independent verification: 核验依据 1 DOI 10.26226/morressier.60546d5e30a2e980041f1f70
#鞅 #可选抽样定理
蒙提霍尔问题（三门问题）里，换门为什么赢的概率更高？
In the Monty Hall problem, why does switching doors win more often?
在 Monty Hall 游戏中，你先选一扇门，主持人（知道门后情况）打开剩下两扇中一扇空门，问你是否换到最后一扇。你最初选中的概率只有 1/3，而奖品在「另两扇门」中的概率是 2/3；主持人排除一扇空门后，这 2/3 全集中到那扇未开的门上，因此换门赢面为 2/3。许多直觉误以为变成 1/2，正是忽略了主持人信息的非随机性。
In the Monty Hall game you pick a door, the host (who knows what's behind) opens an empty one of the other two, and asks if you switch. Your first pick has only a 1/3 chance, while the prize was in the other two doors with probability 2/3; after the host removes an empty door, that 2/3 concentrates on the remaining unopened door, so switching wins 2/3 of the time. The common intuition of 'now 1/2' ignores that the host's reveal is not random.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
《美国统计学家》（The American Statistician，1970s）；大英百科全书（Britannica）
The American Statistician (1970s); Encyclopaedia Britannica
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL16420770W：The new encyclopaedia
#蒙提霍尔 #三门问题 #条件概率 #直觉谬误
辛普森悖论是什么？
What is Simpson's paradox?
辛普森悖论指：各组内都呈现的趋势，在合并数据后却反转。例如两种疗法在各年龄段都更优，但总体却显得更差，原因是各组样本量不均。
Simpson's paradox: a trend appearing in every subgroup reverses when data are pooled, often due to uneven group sizes—classic in medical and social statistics.
难度 进阶 · 适合 高中(15-18)
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)
#概率 #辛普森悖论
什么是样本空间？
What is a sample space?
样本空间是一次随机试验所有可能结果的集合。例如抛一枚硬币的样本空间是 {正面, 反面}。
The sample space is the set of all possible outcomes of an experiment, e.g. {heads, tails} for a coin flip.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #基础
检察官谬误是什么？
What is the prosecutor's fallacy?
它把“若被告无罪，出现该证据的概率很小”错误地等同为“被告无罪的概率很小”。正确的贝叶斯推理还要考虑先验（无辜者基数大得多）。
It confuses P(evidence | innocent) with P(innocent | evidence); Bayes requires also the prior that far more people are innocent than guilty.
难度 进阶 · 适合 高中(15-18)
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.
#概率 #检察官谬误
一种病发病率只有 1%，检测准确率高达 99%，测出阳性就真得病了吗？
A disease affects 1% of people and the test is 99% accurate — does a positive result mean you are almost surely sick?
不一定，这正是贝叶斯定理的用武之地。10000 人里约 100 个病人，其中约 99 个测出阳性；而约 9900 个健康人里有约 99 个被误报为阳性。阳性者共约 198 人，真病人只占一半，所以得病概率约 50%，远不是 99%。先验概率极小的时候，假阳性会淹没真阳性。
Not necessarily — this is exactly what Bayes' theorem handles. Among 10,000 people, about 100 are sick and ~99 of them test positive; among ~9,900 healthy people, ~99 also test positive falsely. Of ~198 positives only about half are actually sick, so the probability of disease given a positive is roughly 50%, far from 99%. When the base rate is tiny, false positives swamp true ones.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《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）
#贝叶斯定理 #假阳性 #先验概率
大数定律说的是什么？它和「赌徒谬误」有什么关系？
What does the Law of Large Numbers state, and how does it relate to the gambler's fallacy?
大数定律指出：随着独立试验次数增多，样本均值会稳定地趋近于理论期望（例如抛硬币正面频率趋于 1/2）。但它并不意味着「短期偏离会被快速扳平」——这是赌徒谬误。每次抛掷仍独立，过去结果不影响未来；大数定律只保证长期频率收敛，而非短期「补偿」。
The Law of Large Numbers says that as the number of independent trials grows, the sample average converges to the theoretical expectation (e.g., coin-flip heads frequency approaches 1/2). It does NOT mean short-term deviations are quickly 'balanced out'—that is the gambler's fallacy. Each toss stays independent; past outcomes don't affect the future. The law only guarantees long-run convergence, not short-term compensation.
难度 进阶 · 适合 初中(12-15)
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）
#大数定律 #样本均值 #期望 #赌徒谬误
伯特兰箱子悖论是什么？
What is Bertrand's box paradox?
三盒分别为“两金”“两银”“金+银”；随机选盒后抽出一枚金球，问另一枚也是金的概率？答案是 2/3，因为金球更可能来自“两金”盒。它考验条件概率的直觉。
Of three boxes (GG, SS, GS) you draw a gold ball; the chance the other is also gold is 2/3, since a gold ball is more likely from the GG box. A lesson in conditional probability.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #伯特兰箱子
什么是独立事件？
What are independent events?
两个事件独立指一个事件的发生不影响另一个发生的概率。例如连续抛硬币，各次结果互不影响。
Two events are independent if one's occurrence doesn't affect the other's probability, e.g. successive coin flips.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #独立事件
蒙提霍尔问题是什么？
What is the Monty Hall problem?
在三扇门后选一扇、主持人打开一扇空门后，换门获胜的概率会从1/3升到2/3。它反直觉地说明“换”更划算。
In the Monty Hall problem, switching doors after a goat is revealed raises win probability from 1/3 to 2/3.
难度 进阶 · 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
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
#概率 #悖论
方差和标准差有什么关系？
How are variance and standard deviation related?
标准差就是方差的平方根。方差衡量数据与均值的平均平方偏差，开方后单位与数据一致，更便于解释。
Standard deviation is the square root of variance; taking the root restores the original units for easier interpretation.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #离散程度
什么是信息熵？
What is information entropy?
熵衡量随机变量的不确定性或信息量，越不可预测则熵越大。它是信息论与机器学习（如决策树）的核心概念。
Entropy measures uncertainty or information content; higher when outcomes are unpredictable. Core to information theory and ML.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
Cover, T. M. & Thomas, J. A.《Elements of Information Theory》,Wiley
Cover, T. M. & Thomas, J. A., Elements of Information Theory, Wiley
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780471241959 (2006) · 核验依据 2 ISBN 9780665590955 (2002)
#概率 #信息论
预报降水概率30%，结果下雨了，是预报错了吗？
If the rain chance was 30% and it rained, was the forecast wrong?
不是。30%表示在类似天气条件下，约三成会下雨。小概率事件照样可能发生，一次结果不能判定概率预报对错。
No. A 30% chance means rain occurs in about 3 of 10 similar situations. Low-probability events still happen, and a single outcome does not invalidate the probability.
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）
#概率 #预报
蒙特卡洛方法靠什么估算答案？
What does the Monte Carlo method rely on to estimate answers?
它用大量随机抽样来逼近积分、概率或复杂系统的期望，样本越多结果越稳（误差约 1/√N）。常用于难以解析求解的高维问题。
It uses massive random sampling to approximate integrals, probabilities or system expectations; more samples give steadier results (error ~1/√N). It shines in high-dimensional problems lacking analytic solutions.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
美国数学会(AMS)《数值分析》
American Mathematical Society (AMS), Numerical Analysis
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780538733519 (2011) · 核验依据 2 ISBN 9780387987934 (1999)
#蒙特卡洛 #抽样
什么是期望值？
What is expected value?
期望值是随机变量按概率加权的平均结果。例如公平骰子点数的期望是 (1+2+3+4+5+6)/6 = 3.5。
Expected value is the probability-weighted average, e.g. a fair die's expectation is 3.5.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #期望值
为什么真正的随机其实很难，电脑多用的是伪随机？
Why is true randomness hard and computers mostly use pseudo-randomness?
真实随机要靠物理噪声，成本高；电脑用确定性的种子加算法生成看似随机的序列，叫伪随机。它可复现、够用，但严格说可预测。
True randomness needs physical noise and is costly; computers use a deterministic seed and algorithm to generate seemingly random sequences, called pseudo-random. Reproducible and good enough, but in principle predictable.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Menezes、van Oorschot、Vanstone《Handbook of Applied Cryptography》（CRC Press）
Menezes, van Oorschot, Vanstone, Handbook of Applied Cryptography, CRC Press
独立核验依据 · Independent verification: 核验依据 1 ISBN 978-1-4398-2191-6 (CRC Press)
#概率 #伪随机
已经知道一件事发生了，再算另一件事的概率，为什么要重新算？
Once we know one event has happened, why must we recompute the probability of another event?
新信息会缩小我们考虑的范围，概率就要跟着更新。比如从 52 张牌中抽一张，先抽到红牌的概率是 26/52；如果已知抽到的是红牌，再算它是红桃的概率就变成 13/26。写成公式：P(A|B)=P(A∩B)/P(B)，读作“在 B 发生的条件下 A 的概率”。
New information shrinks the set of possibilities we consider, so probabilities must be updated. From a 52-card deck, the chance of drawing a red card is 26/52; but given that the card is red, the chance it is a heart becomes 13/26. In symbols: P(A|B)=P(A∩B)/P(B), the probability of A given B.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-03 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》,Oxford University Press,1996,第三版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 3rd ed., 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
#条件概率 #事件 #概率公式
泊松过程（Poisson process）是怎样把“随机事件随时间随机到达”建模成规律的？
How does a Poisson process model 'random events arriving randomly over time' into a regular pattern?
泊松过程要求事件在互不相交的时间段独立发生，且在短时 Δt 内发生一次的概率约 λΔt、发生多次可忽略。于是到时刻 t 的累计事件数服从泊松分布，相邻到达间隔独立且服从指数分布；它刻画了来电、放射性衰变等“平稳随机流”。
A Poisson process assumes events in disjoint time intervals are independent and that in a short Δt the chance of one event is ≈ λΔt while multiple events are negligible. The count by time t is then Poisson, and inter-arrival gaps are independent exponential—capturing 'steady random streams' like calls or radioactive decays.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Ross, S. M., 《随机过程》(第2版), Wiley
Ross, S. M., 'Stochastic Processes' (2nd ed.), Wiley
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL65235W：Stochastic processes（首版 1
#泊松过程 #指数分布 #到达间隔 #随机流
什么是二项分布？
What is the binomial distribution?
它描述 n 次独立重复试验中成功次数的分布，每次成功概率为 p。例如抛10次硬币出现正面次数的分布。
It models the number of successes in n independent trials each with success probability p, e.g. heads in 10 coin flips.
难度 进阶 · 适合 初中(12-15)
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）
#概率 #分布
中心极限定理是什么？
What is the Central Limit Theorem?
它说大量独立同分布随机变量的平均值近似服从正态分布，无论原分布是什么。这是统计推断与许多自然现象呈钟形的原因。
The average of many independent samples approaches a normal distribution regardless of the original one—why bell curves appear and inference works.
难度 进阶 · 适合 高中(15-18)
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）
#概率 #极限定理
为什么期望可以无条件相加，方差却不行？
Why can expectations always be added but not variances?
期望是线性算子：E(X+Y)=E(X)+E(Y) 对任意随机变量成立，即使 X、Y 高度相关。而方差来自平方展开，Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)，只有在两个变量相互独立（协方差为零）时才能简单相加。这正是“期望可线性相加、方差有条件相加”的原因。
Expectation is a linear operator: E(X+Y)=E(X)+E(Y) holds for any random variables, even highly correlated ones. Variance comes from expanding a square, Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y), so it adds only when the variables are independent (covariance zero). That is why expectations add freely while variances add conditionally.
难度 进阶 · 适合 初中(12-15)
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）
#期望的线性性 #方差 #协方差
生日悖论说的是什么？为什么 23 人就很可能撞生日？
What does the birthday paradox say, and why do 23 people likely share a birthday?
生日悖论指：在一个 23 人的房间里，至少有两人生日相同的概率已超过 50%，远快于直觉预期。其原因是比较的是「所有人两两之间」的配对，23 人可产生约 253 对，出现重复的机会迅速累积。到 70 人时该概率已高于 99%。它常被用于说明哈希碰撞与密码学中的生日攻击。
The birthday paradox: in a room of 23 people, the chance that two share a birthday exceeds 50%, far sooner than intuition expects. The reason is we compare all pairs among people—23 people yield about 253 pairs—so coincidences accumulate rapidly. By 70 people the probability tops 99%. It is used to explain hash collisions and birthday attacks in cryptography.
难度 进阶 · 适合 初中(12-15)
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）
#生日悖论 #概率 #配对 #哈希碰撞
泊松分布适合描述什么样的随机现象？
What kind of random phenomena does the Poisson distribution describe?
泊松分布描述相互独立的稀有事件在固定时间（或空间）内发生的次数，例如每小时打进客服的电话数、放射性衰变的计数。若平均发生 λ 次，则 P(X=k)=λᵏe^(−λ)/k!，且它的期望和方差都等于 λ。当二项分布的 n 很大、p 很小时，它也常作为近似。
The Poisson distribution counts independent rare events in a fixed interval of time or space—such as calls per hour to a service center or radioactive decay counts. With mean λ, P(X=k)=λᵏe^(−λ)/k!, and both its mean and variance equal λ. It also approximates the binomial when n is large and p is small.
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）
#泊松分布 #稀有事件 #二项分布近似
为什么 23 个人中就很可能有两个人生日相同？
Why are two people in a room of 23 likely to share a birthday?
这不是悖论，而是概率直觉的反差。先算反面：所有人都不相同。第 2 人避开第 1 人的概率是 364/365，第 3 人避开前两人的概率是 363/365，依此类推，23 个人乘起来的积约 0.49，所以“至少两人相同”约 51%。人数越多越逼近 1，60 人时已超过 99%。
This is not a paradox but a clash with intuition. Compute the complement: all birthdays distinct. Person 2 avoids person 1 with probability 364/365, person 3 avoids the first two with 363/365, and so on; the product over 23 people is about 0.49, so a match has probability roughly 51%. It climbs toward 1 quickly—over 99% by 60 people.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Grinstead, C. M. 与 Snell, J. L., 《概率导论》, 美国数学会 (AMS)
Grinstead, C. M. & Snell, J. L., 'Introduction to Probability', American Mathematical Society
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780821807491 (1997)
#生日问题 #概率直觉 #组合计数
贝叶斯定理解决了什么类型的推理问题？
What kind of reasoning problem does Bayes' theorem address?
贝叶斯定理给出了在观察到新证据后更新某事件发生概率的公式：P(原因|证据) = P(证据|原因)·P(原因) / P(证据)。它把「先验概率」与「似然」结合得到「后验概率」，是医学检测、垃圾邮件过滤和机器学习中不确定性推理的核心。其思想源自托马斯·贝叶斯 18 世纪的遗稿，由拉普拉斯发扬光大。
Bayes' theorem gives the formula for updating the probability of a cause after observing evidence: P(cause|evidence) = P(evidence|cause)·P(cause) / P(evidence). It combines a prior probability with likelihood to yield a posterior, and is central to medical testing, spam filtering, and machine learning. The idea comes from Thomas Bayes's 18th-century posthumous work, developed by Laplace.
难度 进阶 · 适合 初中(12-15)
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）
#贝叶斯定理 #先验 #后验 #条件概率
轮盘连开十次红色后，下一次开黑色会“更该来”吗？
After ten reds in a row on a roulette wheel, is black “due” to come next?
不会。轮盘每次旋转都是独立事件，前一次结果不会影响后一次，下一次开红的概率仍然不变。把“连续失败后该转运了”当成规律，就是著名的赌徒谬误。它混淆了“整段序列的概率”和“单次事件的概率”。
No. Each spin is independent; previous results do not affect the next one, so the probability of red on the next spin is unchanged. Treating “I am due for a win after a losing streak” as a rule is the famous gambler's fallacy. It confuses the probability of a whole sequence with that of a single event.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#赌徒谬误 #独立事件 #概率
强大数定律与中心极限定理分别描述了独立同分布随机变量和的什么行为?两者在信息层次上有何差异?
What aspects of sums of i.i.d. random variables do the strong law of large numbers and the central limit theorem describe respectively, and how do they differ in informational content?
强大数定律给出几乎必然层面的定性结论:(S_n−nμ)/n → 0;中心极限定理进一步定量刻画波动规模:S_n 围绕均值的涨落按 σ√n 的尺度增长,且 √n(S̄_n−μ) 依分布趋于 N(0,σ²)。前者只说偏差消失,后者给出偏差的分布形状,是对涨落的一阶精确逼近,两者互补而非蕴含。
The strong law gives a qualitative almost-sure statement: (S_n−nμ)/n → 0. The central limit theorem quantifies the fluctuation scale: √n(S̄_n−μ) converges in distribution to a normal law. The former only says deviations vanish; the latter pins down their distributional shape, providing first-order accurate approximation of fluctuations — complementary results, neither implying the other.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
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Durrett《Probability: Theory and Examples》第 2 章大数定律与第 3 章中心极限定理
Rick Durrett, Probability: Theory and Examples, Chapter 2 on SLLN and Chapter 3 on CLT
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL15209593W：Probability: Theory an
#大数定律 #中心极限定理
盒子里有2个红球和1个蓝球，摸到红球的可能性大吗？
A box has 2 red and 1 blue ball; is red more likely?
是的。红球比蓝球多，所以摸到红球的可能性更大，概率是 2/3。
Yes—more red balls means red is more likely, with probability 2/3.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《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）
#概率 #基础
什么是可能性（概率）？
What is probability (likelihood)?
可能性表示某件事发生的机会大小，介于0（不可能）和1（必然）之间。
Probability measures how likely an event is, ranging from 0 (impossible) to 1 (certain).
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《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）
#概率 #基础
正态分布（高斯分布）为什么在自然界如此常见？
Why is the normal (Gaussian) distribution so common in nature?
正态分布是呈钟形的对称连续分布，由均值和标准差两个参数决定。它常见的原因在于中心极限定理：许多独立微小随机因素叠加的结果往往趋近钟形。从人的身高到测量误差，大量现象都大致服从它，因此高斯分布成为统计建模与质量控制的基本假设。
The normal distribution is a symmetric bell-shaped continuous distribution determined by just its mean and standard deviation. Its ubiquity follows from the Central Limit Theorem: sums of many independent tiny random factors tend toward a bell shape. From human heights to measurement errors, countless phenomena are approximately normal, making it the default assumption in statistics and quality control.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#正态分布 #高斯 #钟形曲线 #中心极限
为什么掷两颗公平骰子，点数之和为 7 比之和为 2 更可能出现？
Why is the sum 7 more likely than the sum 2 when rolling two fair dice?
两只骰子各有 6 面，共 36 种等可能结果；和为 2 只有 (1,1) 一种，概率 1/36，而和为 7 有 (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) 六种，概率 6/36 = 1/6。组合数越多越可能，这正是两颗骰子点数之和呈现“中间高、两端低”的对称三角形分布的原因。
Two dice give 36 equally likely outcomes; sum 2 occurs only as (1,1), probability 1/36, while sum 7 occurs as six pairs (1,6)…(6,1), probability 6/36 = 1/6. More combinations mean higher probability—the sum of two fair dice follows a symmetric triangular distribution that peaks in the middle.
难度 入门 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-10-04
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Grinstead, C. M. 与 Snell, J. L., 《概率导论》, 美国数学会 (AMS)
Grinstead, C. M. & Snell, J. L., 'Introduction to Probability', American Mathematical Society
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780821807491 (1997)
#骰子 #两骰子求和 #组合数 #概率
抛一枚均匀硬币，正面朝上的概率是多少？
What is the probability of heads when flipping a fair coin?
均匀硬币只有正反两面且机会相等，所以正面朝上的概率是 1/2（也就是一半）。
A fair coin has two equally likely outcomes, so heads has probability 1/2.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《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）
#概率 #基础
什么是赌徒谬误？
What is the gambler's fallacy?
赌徒谬误是误以为独立随机序列会“自我纠正”：连出多次红后觉得黑更可能。其实每次抛硬币独立，过去结果不影响未来，概率始终是 1/2。
The gambler's fallacy wrongly expects independent random events to 'balance out'—after many reds, black seems due. But each fair coin flip stays 1/2 regardless of history.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#概率 #赌徒谬误
买彩票中头奖的概率大概有多小？
How tiny is the chance of winning the top lottery prize?
以双色球为例，头奖概率约为一千七百七十二万分之一。这比被雷击中的概率还小得多，买彩票更应看作娱乐而非投资。
In a typical 6/49-style game the jackpot odds are about 1 in 17.7 million. That is far rarer than being struck by lightning, so treat lottery play as entertainment, not investment.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《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）
#概率 #彩票
抛硬币正面概率是1/2，连抛五次都正面后，下一次更可能是反面吗？
After five heads in a row, is the next coin toss more likely to be tails?
不会。若硬币公平且各次独立，第六次正面的概率仍是1/2。‘该出反面了’是赌徒谬误；过去结果不影响未来的独立事件。
No. If the coin is fair and tosses independent, the sixth toss is still 1/2 heads. 'It's due for tails' is the gambler's fallacy; past results don't affect independent future events.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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 #独立
一个房间只要有23个人,就有超过一半概率存在两人同一天生日?这怎么可能?
Just twenty-three people make shared birthdays likelier than not? How is that possible?
直觉错在把问题当成有人和我同天生日(那确实需要253人);实际问的是任意两人配对,23人共有253对,每对撞日的独立概率约1/365,全部错开的概率连乘后只剩一半以下。这就是著名的生日悖论——并非逻辑矛盾,而是直觉对组合爆炸的低估。它的现实版本威力惊人:密码学中哈希碰撞攻击正基于同一原理,n位指纹只需约2的n/2次方次尝试就能找到碰撞,因此安全系统被迫使用更长的摘要。
Intuition misfires by reading the question as someone sharing YOUR birthday (needing 253 people); really any pair may collide, and twenty-three people form 253 pairs, each matching at about 1/365 independently — multiplying no-collision chances drops below half. The birthday paradox is thus no logical contradiction but intuition underestimating combinatorial explosion. Its practical twin bites cryptography: hash-collision attacks exploit the same math, needing only about two to the n-over-2 tries against n-bit fingerprints — which is why secure systems demand longer digests.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 3 条独立证据
来源 · Source
Feller,《An Introduction to Probability Theory and Its Applications》第3版,Wiley,1968
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, 1968
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)
#生日悖论 #组合概率 #哈希碰撞