MKLab - PROBABILITY AND STATISTICS

A 250-Year-Old Theorem Walks into a Courtroom

Thu, 29 Oct 2026 01:41:20 +0200

A 250-Year-Old Theorem Walks into a Courtroom

In their article, Tommy Andersson and Erik Angner examine how mathematics can help explain the reliability of jury decisions. They discuss research investigating whether increasing the number of jurors improves the probability of reaching a correct verdict. Although a recent study correctly calculated how majority accuracy changes with jury size, the authors point out that its central result had already been established in 1785 by the French mathematician and philosopher Marquis de Condorcet, through what is now known as Condorcet’s Jury Theorem.

The theorem states that when each juror independently has a probability greater than 50% of making the correct decision, the probability that the majority reaches the correct verdict increases as the jury grows. Conversely, if each juror is more likely to be wrong than right, adding more jurors makes the collective decision increasingly likely to be incorrect. The article connects this mathematical result to a much older philosophical debate between Plato, who was sceptical of collective decision-making, and Aristotle, who recognised that groups can benefit from combining the knowledge and judgement of their members.

The authors also explain how the development of probability theory, particularly the work of Jacob Bernoulli and his nephew Nicolaus Bernoulli, made it possible to analyse collective decisions mathematically rather than relying solely on philosophical arguments. The article illustrates the continuing relevance of classical mathematical results to modern research in law, political science and social choice. Its broader lesson is that mathematical knowledge can reveal when collective decisions are more reliable than individual judgements—but only when the underlying assumptions, especially the accuracy and independence of individual decisions, are satisfied.

Source: A 250-Year-Old Theorem Walks into a Courtroom.

Why Probability Theory is Hard

Tue, 29 Sep 2026 00:34:13 +0300

Summary
Probability theory is difficult not because people are unintelligent, but because probability is fundamentally non-intuitive. In everyday life, we learn intuition from systems that behave consistently: if we turn a steering wheel in a particular way, the car responds predictably. Random systems do not behave this way. The same input can produce different outcomes, so the brain cannot easily build the kind of automatic intuition it develops for deterministic systems. As a result, people often look for patterns in randomness that are not actually there.

The second difficulty is that probability generally requires slow, deliberate reasoning rather than instinctive thinking. The article connects this to Daniel Kahneman's distinction between fast, intuitive System 1 thinking and slower, analytical System 2 thinking. Probability problems tend to remain in System 2 because our intuitive mechanisms are poorly adapted to uncertainty. Even extensive experience may improve our ability to solve probability problems, but this often means becoming faster at deliberate reasoning rather than developing a genuinely automatic probabilistic intuition.

Probability is also conceptually confusing because there is no single universally intuitive way to interpret what probability actually represents. Different frameworks—including frequentism, Bayesianism, subjective probability, objective probability, and the Kolmogorov axiomatic approach—emphasize different interpretations. For simple problems such as dice or coin tosses, these distinctions may remain hidden, but they become much more important when dealing with real-world uncertainty. Consequently, students can end up memorizing formulas and rules without developing a deep conceptual understanding.

Nevertheless, the author argues that learning probability is well worth the effort. Even relatively simple probabilistic models can provide a rigorous framework for reasoning about uncertainty and making better-informed decisions. For example, instead of treating an event as simply possible or impossible, probability allows us to quantify uncertainty through values such as $P(A)=0.2$. More sophisticated models can then describe relationships between events using concepts such as conditional probability, $P(A\mid B)$, and expected value, $E[X]$. The central message is therefore that probability is hard because it conflicts with natural intuition—but a relatively small amount of mathematical understanding can go a surprisingly long way.

Key takeaways

Probability is inherently non-intuitive because random systems do not respond predictably to identical conditions.
The human brain is much better at learning deterministic patterns than reasoning naturally about randomness.
Probability requires System 2 thinking: deliberate, analytical, and cognitively demanding reasoning.
Experience can make probability calculations faster and more effective, but does not necessarily create automatic probabilistic intuition.
Probability has several competing interpretations, including frequentist, Bayesian, subjective, objective, and axiomatic approaches.
Students often compensate for the conceptual difficulty by memorizing formulas and procedures rather than understanding the underlying ideas.
Basic probability provides a mathematical framework for dealing with uncertainty, replacing vague intuition with quantitative reasoning.
The difficulty of probability is therefore normal rather than a sign of mathematical inability.
The article's overall message is essentially: probability is hard, but learning even its fundamentals is extremely valuable.

====>ARTICLE<=====

Why Our Brains Do Not Intuitively Grasp Probabilities

Tue, 29 Sep 2026 00:24:03 +0300

Summary
The article explains how randomness and probability can create events that appear to be miracles. For example, two friends may call each other at exactly the same moment, or someone may dream about a person’s death shortly before it happens. Although these events seem extraordinarily unlikely, they become increasingly probable when we consider the enormous number of opportunities for them to occur. Even if an event has a probability of one in a million, millions of opportunities can produce several seemingly miraculous coincidences over time.

The author illustrates this with a calculation based on human perception. Assuming we receive one bit of information per second while awake for 12 hours a day, we process approximately $43,200$ bits of information daily. Even if $99.999%$ of this information is meaningless, the remaining information could still produce approximately $1.3$ seemingly miraculous events per month, or about $15.5$ per year. Confirmation bias then explains why we remember extraordinary coincidences while forgetting the enormous number of ordinary events. We tend to notice evidence that confirms our beliefs and discount evidence that contradicts them.

The article also examines death premonition dreams using probability. An average person experiences approximately five dreams per night, or $1,825$ dreams per year. If only one tenth are remembered, this gives about $182.5$ remembered dreams per person annually. Across $300$ million Americans, this would produce approximately $54.7$ billion remembered dreams per year. Given millions of deaths and billions of social connections between people, it is statistically unsurprising that some dreams will coincidentally correspond to subsequent deaths. In fact, considering the enormous number of possible combinations, it would be surprising if no such coincidences occurred.

Finally, the author introduces the concept of folk numeracy — our natural tendency to misunderstand probability, randomness and statistical trends. Human perception evolved in what Richard Dawkins calls Middle World, the range of sizes, speeds and timescales that humans can directly experience. We easily perceive short-term changes but struggle with extremely large numbers, very slow processes and very long timescales. As a result, we often mistake random fluctuations for meaningful patterns and overlook broader statistical trends. Statistical thinking and probabilistic reasoning can therefore help us distinguish genuine patterns from coincidences.

KEY TAKEAWAYS

Randomness creates coincidences: Extremely unlikely events can occur when there are enough opportunities for them to happen.
Confirmation bias distorts perception: We remember extraordinary coincidences while forgetting countless ordinary events.
Large numbers matter: Billions of events and relationships make seemingly extraordinary coincidences statistically unsurprising.
Anecdotes are not statistical evidence: A coincidence does not by itself establish a causal or supernatural explanation.
Folk numeracy leads to errors: Humans naturally struggle to understand probability, randomness and very large datasets.
Short-term fluctuations can be misleading: We often focus on temporary changes while ignoring long-term trends.
Human perception has limitations: Our brains evolved for everyday scales rather than atomic, cosmic or geological timescales.
Statistical thinking is essential: Probability helps us distinguish meaningful patterns from random events.

ARTICLE

Probability of Randomly Choosing a Prime Number

Sat, 26 Sep 2026 18:13:23 +0300

The concept of probability in the context of choosing a prime number at random relies heavily on the definition of the chosen sample space. A prime number is defined as an integer strictly greater than $1$ that has no positive divisors other than $1$ and itself. When dealing with a finite, bounded set of positive integers—such as the set ranging from $1$ to $n$—the probability $P$ of randomly selecting a prime number is determined by dividing the total count of prime numbers within that range by the overall number of elements $n$. For instance, in the first $10$ positive integers, the prime numbers are $2, 3, 5,$ and $7$, yielding a probability of $P = \frac{4}{10} = 0.4$.

As the upper limit of the sample space $n$ increases, the absolute frequency of prime numbers continues to grow, but the relative density or proportion of these primes begins to decrease. This fundamental behavior is formalized by the famous Prime Number Theorem, which states that the number of prime numbers less than or equal to $n$, denoted as $\pi(n)$, can be approximated as $\pi(n) \approx \frac{n}{\ln n}$. Consequently, the approximate probability of selecting a prime number at random from a large finite set of size $n$ is given by the expression $P \approx \frac{1}{\ln n}$. This inverse logarithmic relationship demonstrates that as numbers become larger, finding a prime becomes significantly less likely.

When considering the asymptotic behavior as the sample space approaches infinity, the limit of this probability tends to zero. Mathematically, this is expressed using limits as $\lim_{n \to \infty} \frac{\pi(n)}{n} = 0$. Even though there are infinitely many prime numbers, their infinite cardinality is sparse compared to the infinitude of all natural numbers. Therefore, if one attempts to choose a prime number uniformly at random from the entire set of infinite integers, the theoretical probability is evaluated as $0$.

In practical applications, such as modern cryptography and number theory, mathematicians frequently test large numbers for primality rather than picking them blindly. Algorithms and theorems—such as Fermat's Little Theorem—help assess whether a given large integer has the characteristics of a prime. By understanding the declining density of primes governed by logarithmic decay, researchers can better estimate the computational effort required to locate large prime numbers essential for secure data encryption protocols.

Key Takeaways

A prime number is an integer greater than $1$ whose only divisors are $1$ and itself.
The basic probability within a finite set is calculated as $P = \frac{\text{Number of Primes}}{\text{Total Numbers in Set}}$.
The Prime Number Theorem estimates the number of primes up to $n$ as $\pi(n) \approx \frac{n}{\ln n}$.
The approximate probability of randomly selecting a prime from numbers up to $n$ is given by $P \approx \frac{1}{\ln n}$.
As the sample space extends to infinity, the limit of the probability drops to zero: $\lim_{n \to \infty} \frac{\pi(n)}{n} = 0$.

ARTICLE / ARTICLE 2

What is the chance of a message in a bottle being found?

Fri, 25 Sep 2026 20:54:13 +0300

The RTÉ Brainstorm article explores the surprisingly small probability that a message in a bottle will both be found and be more than 100 years old. Although bottles can travel enormous distances through ocean currents, many are never found: they may drift into remote regions, break apart, become buried in sediment, or sink after filling with water. The authors, Kevin Burke and David O'Sullivan of the University of Limerick, use this real-world problem to demonstrate how elementary probability can be applied to an unusual historical question.

The first component is the probability that a bottle containing a message is found at all. The authors use an estimate of approximately 1 in 10, based on expert estimates and historical "drift bottle" experiments. Recovery rates have varied considerably by location: some historical experiments produced recovery rates such as 14% in the Gulf of Mexico, 8% in the Caribbean and 7% near northern Brazil, while a more recent Canada–Greenland experiment produced about 5%. Thus, the probability is strongly dependent on geography and ocean circulation.

The second component concerns the age of bottles that are eventually found. Using historical records of old message bottles, the authors observe that the number recovered generally decreases as age increases. They estimate the number of bottles in the 0–25-year category and obtain approximately 106 bottles in their assembled dataset, of which 12 were more than 100 years old. Consequently, the estimated conditional probability that a found bottle is at least 100 years old is approximately $12/106 \approx 1/10$.

The two probabilities can then be combined using the multiplication rule. If $P(F)$ is the probability that a bottle is found and $P(A\mid F)$ is the probability that it is over 100 years old given that it has been found, then $P(F\cap A)=P(F)P(A\mid F)$. Using the article's estimates gives $1/10\times1/10=1/100$. The authors therefore estimate roughly a 1-in-100 chance that a message bottle will both be found and be at least 100 years old. If 100,000 such bottles were floating in the oceans, about 1,000 might satisfy both conditions; under a deliberately simplified assumption that anyone among 8 billion people is equally likely to find one, an individual person's chance would be around 1 in 8 million.

Key takeaways

Multiplication rule:
$P(A\cap B)=P(A)P(B\mid A)$
For the article's estimates:
$P(\text{found})\approx\frac{1}{10}$
Among bottles that are found:
$P(\text{100+ years old}\mid\text{found})\approx\frac{1}{10}$
Therefore:
$P(\text{found and 100+ years old})\approx\frac{1}{10}\times\frac{1}{10}=\frac{1}{100}$
With 100,000 bottles:
$100,000\times\frac{1}{100}=1,000$
With 8 billion people and equal likelihood:
$\frac{1,000}{8,000,000,000}=\frac{1}{8,000,000}$
Important statistical caveat: these are estimates based on incomplete historical data, and the true probability depends heavily on ocean currents, geography, bottle durability, human population density and how likely people are to notice and report a bottle.
Mathematical lesson: a complicated real-world probability can often be decomposed into simpler conditional probabilities and then recombined using the multiplication rule.

ARTICLE

How Random Are Goals in Soccer?

Fri, 25 Sep 2026 20:17:47 +0300

The article “How Random Are Goals in Soccer?” explores whether soccer goals can be treated as random events using statistical analysis. The author examines soccer data from the 2015–16 La Liga season and observes that teams scored an average of approximately $2.74$ goals per match. Although individual goals are influenced by many factors, the article asks whether the aggregate number of goals can nevertheless be described using a relatively simple probability model.

The main mathematical tool is the Poisson distribution. If $X$ represents the number of goals scored in a match and $\lambda$ is the expected number of goals, then the probability of observing exactly $k$ goals is given by $P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}$. Using the observed average, the model takes $\lambda\approx2.74$. This provides a straightforward way of calculating probabilities for outcomes such as 0, 1, 2, 3, or more goals.

The article also considers the timing of goals. If goals were distributed uniformly throughout a 90-minute match, the approximate probability of a goal occurring during any particular minute would be $p\approx\frac{2.74}{90}\approx0.0304$. By simulating many matches using this probability, one can generate a distribution of total goals that resembles the distribution found in real soccer data. This demonstrates an important statistical principle: a process can appear highly unpredictable at the individual level while still exhibiting predictable patterns when observed over many repetitions.

However, soccer is not genuinely a perfectly random Poisson process. The probability of scoring can change depending on the scoreline, the strength of the teams, tactical decisions, substitutions, red cards, fatigue, and the time remaining. Consequently, the Poisson model should be regarded as a useful approximation rather than a complete description of soccer. Its real value is methodological: it demonstrates how relatively simple mathematics, probability theory, and computer simulation can be used to investigate a complex real-world phenomenon.

Key Takeaways

Soccer goals can be approximated using probabilistic models.
The average number of goals per match in the dataset is approximately $\lambda=2.74$.
The Poisson probability formula is $P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}$.
With $\lambda=2.74$, the model can estimate the probability of different goal totals.
An approximate per-minute goal probability is $p=\frac{2.74}{90}\approx0.0304$.
Individual events can be unpredictable while aggregate behaviour remains statistically predictable.
Simulation allows the theoretical probability model to be compared with real soccer data.
The Poisson model assumes events occur independently at a roughly constant rate.
Real soccer violates these assumptions because the scoring rate changes with game conditions.
The example provides a practical introduction to Poisson distributions, expected value, probability, independence, random variables, and Monte Carlo simulation.
For teaching mathematics, soccer provides an intuitive real-world context for explaining why statistical models do not need to reproduce every individual event to describe the overall behaviour of a system.

ARTICLE

Probability underlies much of the modern world

Fri, 25 Sep 2026 20:10:51 +0300

Probability is presented as a mathematical framework for describing randomness and uncertainty, rather than a method for predicting every individual event. A key distinction is between an individual outcome and the long-run patterns that emerge from repeated experiments. For example, a fair coin has probability $P(H)=0.5$ of landing heads, but eight actual flips need not produce exactly four heads. The article emphasizes that probability depends on how we define the sample space, the events being studied, and the assumptions incorporated into our model.

The coin-flipping example illustrates an important principle of mathematical modeling: probability is not necessarily a perfect description of physical reality. Real coins are influenced by factors such as their initial orientation and the mechanics of the toss. Nevertheless, when a physical process is sufficiently sensitive to small variations, treating it probabilistically can provide a useful model. The article also distinguishes probability from frequency: a frequency is what has actually happened in a finite sequence, while probability describes the expected long-run behavior of a model.

The article then connects probability to modern technology. Artificial intelligence, particularly large language models, relies heavily on probabilities when predicting what token or word should come next. Cryptography also depends on randomness: secure passwords and cryptographic keys require outcomes that are difficult for an attacker to predict. In statistics and experimental science, randomization is equally important. Randomized controlled trials use random assignment to reduce systematic differences between treatment groups, helping researchers determine whether an observed effect can be attributed to the treatment rather than to other variables.

The broader message is that probability is best understood as a model for reasoning under uncertainty. It does not eliminate randomness or guarantee that an individual event will conform to the expected pattern. Instead, it allows us to quantify uncertainty and make meaningful statements about collections of events. This makes probability fundamental not only to mathematics and statistics but also to AI, cryptography, scientific experimentation, and many other aspects of modern life.

Key Takeaways

Probability models uncertainty: it assigns numerical values between $0$ and $1$ to possible events.
Probability ≠ frequency: frequency describes observed outcomes; probability describes the behavior expected from a model in the long run.
For a fair coin, $P(H)=P(T)=\frac{1}{2}$, but finite experiments can deviate substantially from $50$–$50$.
The Law of Large Numbers explains why empirical frequencies tend to approach theoretical probabilities as the number of trials increases:
Xnn→pas n→∞.\frac{X_n}{n}\rightarrow p \quad \text{as } n\rightarrow\infty.
Probability depends on the model and its assumptions, including the choice of sample space.
Randomness can be useful rather than merely inconvenient: it is fundamental to cryptography and statistical experimentation.
Modern AI systems use probability to estimate likely next tokens:
P(xt+1∣x1,x2,…,xt).P(x_{t+1}\mid x_1,x_2,\ldots,x_t).
Randomized controlled trials use random assignment to help separate causal effects from confounding factors.
A probability model does not necessarily describe every physical detail; its value lies in capturing the aspects of uncertainty that matter for the problem.
Probability is therefore a central mathematical language for reasoning about an uncertain modern world.

ARTICLE

Cox's theorem

Fri, 25 Sep 2026 16:00:01 +0300

Cox’s theorem, developed by physicist Richard T. Cox, attempts to derive the mathematical laws of probability from general principles governing plausible reasoning under uncertainty. Instead of starting with probability axioms, Cox asks what numerical system should represent how plausible a proposition is, given some information. The central idea is that if plausibility is represented consistently, is comparable on a numerical scale, and behaves sensibly when logical propositions are combined, then the resulting system is essentially probability theory. This gives probability a logical or Bayesian interpretation, where probabilities quantify degrees of belief in propositions rather than merely long-run frequencies.

Cox represents the plausibility of a proposition $A$ given information $X$ by $A\mid X$. For two propositions $A$ and $B$, he assumes that the plausibility of their conjunction can be determined from the plausibilities of its components. This produces a functional equation such as $AB\mid X=g(A\mid X,B\mid AX)$. Consistency with the associativity of logical conjunction forces $g$ to have an associative structure. Under the relevant monotonicity assumptions, a suitable transformation $w$ converts this operation into multiplication: $w(AB\mid X)=w(A\mid X)w(B\mid AX)$. Thus, an abstract system of plausible reasoning begins to acquire the familiar mathematical structure of conditional probability.

The treatment of negation supplies the second major ingredient. Cox assumes that the plausibility of $\neg A$ is determined by the plausibility of $A$, through a function $f$: $w(\neg A\mid X)=f(w(A\mid X))$. Because applying negation twice returns the original proposition, $f$ must satisfy $f(f(x))=x$. Combining the requirements for conjunction and negation ultimately gives a probability-like representation. In particular, there exists a positive number $m$ such that $w^m$ satisfies $w^m(A\mid B)+w^m(\neg A\mid B)=1$ and $w^m(AB\mid C)=w^m(A\mid C)w^m(B\mid AC)$. Defining $P=w^m$ produces the familiar probability rules.

The theorem is important because it provides one mathematical justification for Bayesian probability as a logic of uncertain reasoning. However, it is not an unconditional proof that probability is the only possible framework for uncertainty. The conclusion depends on Cox's assumptions, and later authors have questioned or modified some of them. In particular, Halpern identified counterexamples under weakened assumptions, while other researchers have proposed alternative formulations or additional assumptions. Another important technical point is that Cox's derivation gives finite additivity, whereas the standard measure-theoretic formulation of probability uses the stronger property of countable additivity.

Key takeaways

Main idea: consistent numerical reasoning about uncertain propositions leads to the mathematical structure of probability.
Starting point: assign a real-valued plausibility to propositions rather than assuming probability axioms from the beginning.
Conjunction: consistency leads to a multiplicative rule, ultimately giving
$P(A\cap B\mid C)=P(A\mid C)P(B\mid A\cap C)$.
Negation: the complement rule becomes
$P(\neg A\mid B)=1-P(A\mid B)$.
Logical structure: associativity of conjunction and the behavior of negation generate functional equations whose solutions have probability-like form.
Bayesian interpretation: probability can be viewed as an extension of ordinary logic to situations involving uncertainty.
Important limitation: Cox's result depends on mathematical assumptions about how plausibility behaves; it does not establish that every conceivable theory of uncertainty must be ordinary probability.
Measure-theoretic distinction: Cox's basic derivation yields finite additivity, while Kolmogorov's standard framework requires countable additivity.
Mathematical interest: the theorem is closely connected with functional equations, associativity, monotonicity, Boolean logic, and probability theory.
Historical significance: Cox's original work appeared in 1946, with a more extensive treatment published in 1961.

ARTICLE / PDF

Topology and geometry in data science

Tue, 15 Sep 2026 19:03:26 +0300

The article “On the shape that matters – topology and geometry in data science” by Paweł Dłotko explores how topological data analysis (TDA) can capture the shape and structure of datasets in ways that traditional statistics often cannot. While conventional statistical methods reduce complex, high-dimensional data to numerical summaries, datasets with almost identical statistics can have radically different geometric shapes.

TDA addresses this by constructing mathematical objects such as simplicial complexes from point clouds and studying their topological properties—connected components, holes, and cavities—as the scale of observation changes. The article introduces constructions such as Vietoris–Rips and alpha complexes, which turn raw data into multiscale structures known as filtrations.

A central tool discussed is the Euler characteristic curve (ECC), which summarizes how the topology of a dataset evolves across different scales. The ECC can provide robust descriptions of high-dimensional data because it is invariant under transformations such as permutations of coordinate axes and affine transformations. The article particularly highlights its use in statistical goodness-of-fit testing: the resulting “TopoTests” use topological information as an alternative to the traditional cumulative distribution function and can outperform established methods, even for relatively small datasets. Overall, the article presents topology and geometry not merely as abstract mathematical theories, but as practical tools for visualising, comparing, classifying and statistically analysing complex data, with applications ranging from data science to cosmology.

ARTICLE [PDF]

Where Did Combinators Come From ?

Sun, 06 Sep 2026 00:28:45 +0300

Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel
Author: Stephen Wolfram
Submitted: 18 August 2021
Field: History of Mathematics / Mathematical Logic / Foundations of Computation
arXiv: 2108.08707 (arXiv)

Stephen Wolfram’s article is a historical investigation of Moses Schönfinkel (1888–?), the mathematician who introduced combinators in a lecture to the Göttingen Mathematical Society on 7 December 1920. Wolfram argues that Schönfinkel’s work deserves a much more prominent place in the history of computation: sixteen years before Turing machines, Schönfinkel developed a formal system in which arbitrary functions could, in principle, be represented by combinations of a tiny number of primitive operations. His ideas appeared in the 1924 paper Über die Bausteine der mathematischen Logik (“On the Building Blocks of Mathematical Logic”). The essential primitives are what are now called the $S$ and $K$ combinators, which may be written as
$Kxy=x$
and
$Sxyz=xz(yz)$.

From combinations of these extraordinarily simple rules one can construct general computations without introducing explicitly named variables. Wolfram therefore characterizes Schönfinkel’s system as an exceptionally early—and strikingly minimal—formalism for what we now understand as universal computation.

Much of the paper reconstructs Schönfinkel’s almost-lost biography from archival material. Born in Ekaterinoslav, now Dnipro, in 1888, Schönfinkel studied mathematics at Odessa and later went to Göttingen, where he was connected with the intellectual circle surrounding David Hilbert and Paul Bernays. He helped prepare notes for Hilbert’s 1920 lectures on mathematical logic and presented his revolutionary combinator ideas later that year. Yet his academic career never developed conventionally. In 1924 he left Göttingen for Moscow; after that, reliable information becomes scarce. Contemporary reports claimed that he became mentally ill and entered a sanatorium, while later stories said that he died impoverished around 1940–42, but Wolfram emphasizes that much of this remains uncertain and is supported largely by indirect testimony rather than definitive records.

The larger theme is how an extraordinarily important idea can almost disappear from intellectual history. Haskell Curry encountered Schönfinkel’s work in 1927 and subsequently developed combinatory logic much further, while lambda calculus and Turing machines eventually became much better-known foundations for computation. Schönfinkel’s original paper largely vanished from attention until it was republished in English in Jean van Heijenoort’s influential 1967 collection From Frege to Gödel. Wolfram argues that Schönfinkel’s achievement was especially remarkable because he identified the essential structure of computation before computers existed and before “computation” itself had become a clearly articulated mathematical concept. In this sense, the whole intellectual legacy of an almost forgotten mathematician became compressed into two tiny symbols: $S$ and $K$.

Key takeaways

Schönfinkel anticipated fundamental ideas of theoretical computer science in 1920, long before Turing machines and modern programming languages.
The combinators $S$ and $K$ show that extremely simple symbolic transformation rules can generate arbitrarily complex computations.
His work became the foundation of combinatory logic and is closely related to lambda calculus, functional programming, and ideas such as currying.
Wolfram’s paper is as much a piece of mathematical detective work as mathematics: it reconstructs the life of a nearly forgotten figure whose ideas became fundamental to our modern conception of computation.

ARTICLE [PDF]

History as a giant data set

Sat, 05 Sep 2026 23:47:08 +0300

History as a Giant Data Set: How Analysing the Past Could Help Save the Future
Author: Laura Spinney
Publication:The Guardian
Publication date: 12 November 2019

The article explores cliodynamics, an emerging discipline that treats history as a vast data set and uses mathematics, statistics and computer modelling to search for recurring patterns in the rise and decline of societies. Its central figure is Peter Turchin, originally a mathematical biologist, who argues that social systems can be studied much like ecological systems. His models suggest that societies often experience long “secular cycles” of prosperity and instability, driven by interacting factors such as population pressure, declining living standards, competition among elites, inequality and weakening state finances. These ideas build substantially on the earlier work of Jack Goldstone, who developed a political stress indicator, usually denoted $\Psi$, to represent the combined pressures generated by mass mobilisation, elite competition and fiscal weakness.

A major development behind this approach is the availability of enormous historical databases such as Seshat, which collects quantitative information about hundreds of societies across thousands of years. Researchers can use indirect evidence—coin hoards, skeletal remains, building sizes, wages, demographic records and even environmental proxies—to reconstruct variables that cannot be measured directly. Turchin argues that such data reveal recurring long-term cycles as well as shorter roughly 50-year “fathers-and-sons” oscillations in political violence. Applied to the United States, his models showed rising structural political stress from around the 1970s onward, which led him years in advance to identify the period around 2020 as one of elevated instability.

The article nevertheless emphasizes that cliodynamics is probabilistic rather than deterministic. It cannot predict the exact event that triggers a revolution or crisis; instead, it attempts to measure whether a society has become structurally vulnerable, in much the same way that meteorology estimates the probability of severe weather. Critics—particularly many traditional historians—argue that human behaviour is too contingent and culturally specific to be captured reliably by equations, and that large historical databases may reproduce biases or strip evidence of its context. Turchin's more ambitious claim is therefore not that mathematics can foretell history, but that quantitative models might eventually function as an early-warning system for societies, identifying dangerous structural pressures and allowing governments to intervene before instability turns into collapse.

Key takeaways

History may contain measurable recurring patterns, rather than being entirely a sequence of unique events.
Cliodynamics combines history + mathematics + statistics + complexity science + big data.
Social instability tends to rise when inequality, elite competition, popular hardship and state weakness reinforce one another.
The goal is not exact prophecy, but something closer to a “weather forecast” for political and social instability.
The deepest question raised by the article is whether understanding historical cycles can help societies change their trajectory rather than simply repeat them.

ARTICLE / ARTICLE [PDF]

Bayes theorem

Fri, 04 Sep 2026 04:24:41 +0300

Summary

Bayes’ theorem is a fundamental result in probability theory that describes how to update the probability of a hypothesis when new evidence becomes available. If $A$ represents a hypothesis and $B$ represents observed evidence, then
$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$
where $P(A)$ is the prior probability of $A$, $P(B\mid A)$ is the likelihood of observing $B$ assuming $A$ is true, and $P(A\mid B)$ is the posterior probability after incorporating the evidence. The theorem follows directly from the definition of conditional probability and the identity
$P(A\cap B)=P(A\mid B)P(B)=P(B\mid A)P(A)$.

The importance of Bayes’ theorem lies in its ability to reverse conditional probabilities. Knowing, for example, how likely a positive medical test is when a patient has a disease does not immediately tell us how likely the patient is to have the disease after testing positive. Bayes’ theorem combines the test's accuracy with the disease's underlying prevalence to obtain that probability. This distinction is particularly important when dealing with rare events, because even a highly accurate test can produce a substantial proportion of false positives when the prior probability of the condition is very small.

The theorem is named after Thomas Bayes, whose work on conditional probability was published posthumously in 1763, although Pierre-Simon Laplace independently developed and greatly extended the approach. Today Bayes’ theorem forms the mathematical foundation of Bayesian inference, where beliefs about unknown parameters or competing hypotheses are repeatedly updated as new data arrive. Its applications extend across statistics, machine learning, medical diagnosis, genetics, cryptanalysis, decision theory, and many other areas involving reasoning under uncertainty.

Key takeaways

Prior + evidence → posterior: Bayes’ theorem provides a systematic mathematical rule for updating probabilities.
$P(A\mid B)\neq P(B\mid A)$: confusing these two conditional probabilities is a common error.
Base rates matter: rare events can remain relatively unlikely even after apparently strong evidence.
Bayes’ theorem is one of the central mathematical tools behind Bayesian statistics and probabilistic machine learning.

ARTICLE / ARTICLE [PDF]

Gambler's fallacy

Thu, 03 Sep 2026 02:14:35 +0300

Summary

The gambler’s fallacy is the mistaken belief that past outcomes of independent random events influence what will happen next. For example, after a fair coin lands heads several times in a row, someone may believe that tails is now “due.” In reality, each toss is independent, so the probability remains
$P(heads)=P(tails)=12.P(\text{heads})=P(\text{tails})=\frac12$.
Likewise, if four heads have already occurred, the probability that the fifth toss is heads is still $1/2$. The confusion comes from mixing up the probability of predicting an entire sequence beforehand with the probability of the next event after part of the sequence has already occurred. Five consecutive heads have probability $1/2^5=1/32$ before the experiment begins, but once four heads have already occurred, the probability of another head is simply $1/2$.

The fallacy is also called the Monte Carlo fallacy, after a famous roulette event at the Monte Carlo Casino on August 18, 1913. The ball reportedly landed on black 26 times consecutively, leading gamblers to bet increasingly large sums on red because they believed red had to appear soon. But if the roulette wheel was unbiased, each new spin remained independent of the preceding spins. The psychological explanation is closely associated with what Kahneman and Tversky called the representativeness heuristic: people expect even short random sequences to look balanced, although genuine randomness naturally produces streaks and clusters.

Importantly, the gambler’s fallacy applies only when events really are independent and probabilities remain unchanged. Drawing cards without replacement is different: after an ace is removed from a deck, the probability of drawing another ace genuinely changes. Similarly, an extraordinary run of identical results might provide evidence that a coin, roulette wheel, or other mechanism is biased; in that case, updating probabilities based on the observations can be rational rather than fallacious.

Key takeaways

Independent random events do not remember previous outcomes.
A long streak does not make the opposite outcome “due.”
Random sequences naturally contain surprisingly long streaks.
The law of large numbers does not require short-term imbalances to be immediately corrected.
Past outcomes matter when events are dependent, probabilities change, or observations provide evidence that the underlying process is biased.

ARTICLE

Bell number

Tue, 01 Sep 2026 23:55:14 +0300

Bell Numbers — Summary

The Bell numbers $B_n$ form an important sequence in combinatorics. They count the number of ways an $n$-element set can be divided into nonempty, disjoint subsets. In other words, $B_n$ gives the total number of possible partitions of a set, where the order of the subsets does not matter.
The sequence begins
$B_0=1,\quad B_1=1,\quad B_2=2,\quad B_3=5,\quad B_4=15,\quad B_5=52,\quad B_6=203,\ldots$
For example, the set ${a,b,c}$ has exactly five possible partitions, so $B_3=5$. Bell numbers also count the number of possible equivalence relations on a set containing $n$ elements.

Bell numbers are closely connected with the Stirling numbers of the second kind. If $S(n,k)$ denotes the number of ways to partition a set of $n$ elements into exactly $k$ nonempty subsets, then
$B_n=\sum_{k=0}^{n}S(n,k)$.
They also satisfy the recurrence relation
$B_{n+1}=\sum_{k=0}^{n}\binom{n}{k}B_k$.
Their exponential generating function is
$\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n=e^{e^x-1}$.

These formulas show the close relationship between Bell numbers, set partitions, Stirling numbers and generating functions. Bell numbers grow very rapidly as $n$ increases, and their asymptotic behaviour can also be described using the Lambert $W$ function.
The numbers are named after the mathematician Eric Temple Bell, who studied them during the 1930s, although the underlying combinatorial ideas appeared much earlier.
An interesting historical example comes from Japan. A traditional incense game associated with The Tale of Genji involves five groups and produces exactly $B_5=52$ possible patterns, corresponding to the 52 possible partitions of a five-element set.

Key takeaways

Definition: $B_n$ is the number of partitions of an $n$-element set.
First values: $1,1,2,5,15,52,203,877,4140,\ldots$
Connection with Stirling numbers: $B_n=\sum_{k=0}^{n}S(n,k)$.
Recurrence relation: $B_{n+1}=\sum_{k=0}^{n}\binom{n}{k}B_k$.
Generating function: $\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n=e^{e^x-1}$.
Area of mathematics: mainly combinatorics and discrete mathematics, with connections to probability, number theory and analytic combinatorics.

ARTICLE

Benford's law

Sun, 23 Aug 2026 23:18:13 +0300

Benford’s Law
The Strange Mathematics of First Digits

Benford’s law, also known as the Newcomb–Benford law or the first-digit law, is a remarkable mathematical phenomenon concerning the frequency with which digits appear at the beginning of numbers in many real-world datasets.
One might naturally expect the digits 1 through 9 to appear as the first digit with approximately equal probability. If that were true, each digit would occur about $11.1%$ of the time.
Surprisingly, this is often not what happens.
In many naturally occurring collections of numbers, values beginning with 1 appear much more frequently than values beginning with 9.

The Law
According to Benford’s law, the probability that the first significant digit of a number is $d$ is

$P(d)=\log_{10}\left(1+\frac{1}{d}\right)$

where

$d=1,2,\ldots,9$

For example,

$P(1)=\log_{10}(2)\approx0.301$

so approximately 30.1% of the numbers in a Benford-distributed dataset begin with the digit 1.
For the digit 9,

$P(9)=\log_{10}\left(\frac{10}{9}\right)\approx0.046$

so only about 4.6% begin with 9.
This means that numbers beginning with 1 can occur roughly six times as often as numbers beginning with 9.

First-Digit Probabilities
1 → 30.1%
2 → 17.6%
3 → 12.5%
4 → 9.7%
5 → 7.9%
6 → 6.7%
7 → 5.8%
8 → 5.1%
9 → 4.6%
The probability steadily decreases as the leading digit becomes larger.

Why Does This Happen?
The explanation is closely related to logarithmic scales.
On a logarithmic scale, the interval occupied by numbers beginning with 1 is considerably larger than the interval occupied by numbers beginning with 9.
For example, numbers with leading digit 1 occupy the logarithmic interval from

$\log_{10}(1)$ to $\log_{10}(2)$

whose length is

$\log_{10}(2)-\log_{10}(1)=\log_{10}(2)\approx0.301.$

Numbers beginning with 9 occupy only the interval from

$\log_{10}(9)$ to $\log_{10}(10)$

whose length is approximately

$0.046.$

Thus, if the logarithms of the values are distributed approximately uniformly, Benford’s law appears naturally.

Where Does Benford’s Law Appear?
The law tends to work particularly well for datasets that span several orders of magnitude.
Examples can include:
Population figures
Financial and accounting data
Company revenues
Geographical measurements
Physical quantities
Economic statistics
Scientific measurements
Certain mathematical sequences
Interestingly, some purely mathematical sequences also approach Benford’s distribution.
Examples include the Fibonacci numbers, factorials and powers of 2.

A Historical Curiosity
The phenomenon was first noticed by the astronomer and mathematician Simon Newcomb in 1881.
Newcomb noticed something peculiar about books containing logarithm tables.
The pages containing logarithms of numbers beginning with small digits—especially 1—were considerably more worn than pages corresponding to numbers beginning with larger digits.
He reasoned that people must therefore be looking up numbers beginning with 1 much more frequently than numbers beginning with 9.
The observation was largely forgotten.
More than fifty years later, physicist Frank Benford independently rediscovered the phenomenon.
In 1938, Benford published a large investigation involving more than 20,000 numerical observations taken from many different sources.
The phenomenon subsequently became known as Benford’s law.

Fraud Detection
One of the most fascinating modern applications of Benford’s law is in forensic accounting and fraud detection.
Suppose that a large collection of financial transactions would normally be expected to follow Benford’s law.
An auditor can compare the actual frequencies of the first digits with the theoretical frequencies predicted by Benford’s distribution.
Large or systematic deviations may identify data that deserve further investigation.
For this reason, Benford analysis has been used in areas such as:
Accounting audits
Tax investigations
Financial fraud detection
Economic statistics
Scientific datasets
Election-data analysis
However, there is an extremely important limitation.
Failure to follow Benford’s law is not proof of fraud.
It merely indicates that the dataset may deserve closer examination.

When Benford’s Law Does NOT Apply
Not every collection of numbers should follow Benford’s law.
It generally performs poorly when numbers are artificially restricted to a narrow interval or assigned according to human-designed systems.
Examples include:
Human heights
IQ scores
Telephone numbers
Postal codes
Identification numbers
Sequential invoice numbers
Numbers with predetermined minimum or maximum values
For example, the heights of adults might mostly lie between roughly 150 and 200 centimetres. Such values do not span several orders of magnitude, so there is no reason to expect them to follow Benford’s law.
This limitation is particularly important when Benford analysis is used to make claims about fraud or manipulation.

Why Is Benford’s Law So Interesting?
Benford’s law is fascinating because it demonstrates that something as apparently arbitrary as the first digit of a number can exhibit a strong mathematical pattern.
It connects several areas of mathematics and statistics, including:
Probability theory
Mathematical statistics
Number theory
Logarithms
Data analysis
Forensic statistics
Perhaps most surprisingly, the law shows that in many datasets the digits 1 through 9 are far from equally likely.
The humble digit 1 appears at the beginning of numbers approximately one-third of the time.

Key Takeaways
1. In many naturally generated datasets, first digits are not uniformly distributed.
2. About 30.1% of Benford-distributed numbers begin with 1.
3. Only about 4.6% begin with 9.
4. The distribution is described by

$P(d)=\log_{10}\left(1+\frac{1}{d}\right).$

5. The phenomenon is fundamentally connected with logarithmic scales.
6. Benford’s law is useful for detecting statistical anomalies, particularly in financial data.
7. A deviation from Benford’s law is not, by itself, evidence of fraud.

Source:
ARTICLE / ARTICLE [PDF]