Mathematics & Probability

Bayes Theorem

Bayes / Price / Laplace / Fisher

Update your beliefs proportionally to how surprising the new evidence is — the foundation of rational inference.

Black Swan Theory

Nassim Nicholas Taleb

Rare, unpredictable events with extreme impact shape history.

Compounding

Einstein

Exponential growth from consistent small gains over long time horizons.

Correlation vs Causation

Moving together does not mean one causes the other.

Ergodicity

What works across many simultaneously may not work for one across time.

Exponential Growth

Growth accelerates proportionally to current size, slow then explosive.

Fermi Problem

Enrico Fermi

Break impossible questions into tractable sub-problems for useful estimates.

Global & Local Maxima

The best nearby outcome may trap you from reaching the best possible outcome.

Inversion

Carl Jacobi / Charlie Munger

Think backwards: instead of asking how to succeed, ask what guarantees failure — then avoid it.

Kelly Criterion

John Kelly Jr.

Optimal bet sizing maximizes long-term wealth growth while avoiding ruin.

Nonlinearity

Outputs not proportional to inputs; small changes can produce massive effects.

Power Law Distribution

Vilfredo Pareto

A small number of items account for a disproportionate share of outcomes.

Probabilistic Thinking

Thomas Bayes / Nate Silver

Replace binary judgments with probability estimates under uncertainty.

Probability Theory

Pascal / Fermat / Kolmogorov

The mathematical framework for quantifying uncertainty.

Regression to the Mean

Francis Galton

Extreme outcomes are followed by more moderate ones as luck fades.

Signal vs Noise

Claude Shannon

Distinguish meaningful information from irrelevant data.

Algorithms

An algorithm is a finite, unambiguous sequence of steps that transforms inputs into outputs. No ambiguity, no infinite loops in the intended design — given the same input, the...

Anti-patterns

An anti-pattern is a recurring solution that looks plausible but backfires: it solves a surface problem while creating worse ones or locking in failure. The term came from...

Apophenia

Klaus Conrad

Apophenia is the tendency to perceive meaningful patterns in random or meaningless data. The brain is a pattern-seeking engine; when it runs on noise, it still finds structure....

Back-of-envelope Calculation

A back-of-envelope calculation is a quick, approximate computation done with minimal tools — napkin, envelope, or mental math — to bound a problem, check an claim, or decide...

Central Limit Theorem

The Central Limit Theorem (CLT) says that the distribution of the sum (or average) of many independent random variables tends toward a normal distribution, regardless of the shape...

Conditional Probability

Conditional probability is the probability of an event given that another event has occurred: P(A|B) = P(A and B) / P(B). It is the workhorse of updating beliefs with evidence....

Confounding Factor

A confounding factor (confounder) is a variable that influences both the supposed cause and the supposed effect, creating a spurious association between them. You see that X and Y...

Gambler's Ruin

Pascal / Huygens

Gambler's ruin is the mathematical certainty that a player with finite capital playing against a house (or opponent) with effectively infinite capital will go broke with...

Inflection Point

Andy Grove

An inflection point is the moment when the rate of change itself changes — the curve stops bending one way and starts bending the other. In calculus, it's where the second...

J-Curve

The J-curve is the shape of outcomes when things get worse before they get better. Performance drops, then recovers and can exceed the starting point — the path traces a "J" when...

Law of Large Numbers

Jacob Bernoulli / Andrey Kolmogorov

The law of large numbers says that as you repeat a random experiment more and more times, the average of the results converges to the expected value. Sample a few times and the...

Law of Small Numbers

Kahneman & Tversky

The law of small numbers is a cognitive bias: we treat small samples as if they were large. We see patterns, stability, and representativeness in a handful of observations where...

Long Tail

Chris Anderson

The long tail is the large number of low-volume items that collectively can match or exceed the sales of the few blockbusters. In a traditional store, shelf space is scarce; you...

Major vs Minor Factors

Major vs minor factors is the discipline of separating what actually moves the outcome from what doesn't. Most outcomes are driven by a few variables; the rest are noise or...

Monte Carlo Fallacy

The Monte Carlo fallacy — also called the gambler's fallacy — is the belief that past independent outcomes change the probability of future ones. A coin that landed heads five...

Monte Carlo Simulation

Stanisław Ulam / John von Neumann

Monte Carlo simulation estimates outcomes by running many random trials instead of solving equations. You define a model (inputs, relationships, and distributions), draw random...

Multiplying by Zero

In a chain of multiplicative factors, one zero makes the product zero. Revenue = traffic × conversion × price × retention. If conversion is zero, revenue is zero no matter how...

Order of Magnitude

Order of magnitude is the size of something expressed as a power of ten. Is it 10, 100, 1,000, or 10,000? Getting the order of magnitude right means you are in the right ballpark...

Outliers

Outliers are observations that lie far from the bulk of the data — the extreme values that don't fit the typical pattern. How you treat them shapes understanding and decisions....

Queuing Theory

Queuing theory is the study of waiting lines: how jobs, requests, or customers arrive, how they are served, and how long they wait. The core result is that wait times and queue...

Randomness

Randomness is the absence of a deterministic pattern: outcomes that cannot be predicted with certainty from prior information. In practice, we treat a process as random when we...

Reference Class Forecasting

Kahneman & Tversky / Bent Flyvbjerg

Reference class forecasting answers a single question: when projects like this one were attempted, what actually happened? Instead of building a bottom-up estimate from tasks and...

Scale

Scale is the relationship between size and outcome: how do cost, performance, or value change as you move from one magnitude to another? The question is not "is bigger better?"...

Scenario Analysis

Pierre Wack

Scenario analysis replaces a single forecast with a small set of distinct futures — typically base, upside, and downside — and asks what happens in each. The goal is not to assign...

Scope Neglect

Paul Slovic

Scope neglect is the failure to adjust judgment or willingness to pay when the scale of the problem changes. People often react similarly to "100 birds at risk" and "10,000 birds...

Sensitivity Analysis

Sensitivity analysis asks: when we change one input, how much does the output change? It identifies which assumptions drive the result. If a 10% drop in conversion rate flips your...

Simpson's Paradox

Edward Simpson / Yule

Simpson's paradox is the reversal of a trend or comparison when data are aggregated versus when they are split into groups. A treatment can look better in every subgroup but worse...

Standard Deviation & Normal Distribution

Standard deviation measures spread: how far outcomes typically are from the average. If the mean is 100 and the standard deviation is 15, then roughly two-thirds of observations...

Algebra

Algebra is the practice of using symbols to represent unknown quantities and relationships, then manipulating those symbols to solve for unknowns. As a mental model, its power...

Anscombe's Quartet

Francis Anscombe

Anscombe's Quartet is a set of four datasets that share nearly identical summary statistics — same mean, variance, correlation, and regression line — yet look completely different...

Benford's Law

Benford's Law states that in many naturally occurring datasets, the leading digit is not uniformly distributed — the digit 1 appears as the first digit about 30% of the time,...

Berkson's Paradox

Berkson

Berkson's Paradox is a statistical phenomenon where two variables that are unrelated — or even positively correlated — in the general population appear negatively correlated when...

Confidence Intervals

A confidence interval expresses the range within which a true value likely falls, given the uncertainty in your measurement. A 95% confidence interval means that if you repeated...

Data Dredging

Data dredging — also called data fishing or data snooping — is the practice of searching through large datasets for statistically significant patterns without a prior hypothesis....

Dimensional Analysis

Dimensional analysis checks whether an equation or estimate makes sense by verifying that the units on both sides match. If you're calculating cost per customer and your answer...

Equivalence

Equivalence is the recognition that two things different in form are identical in function, value, or logical structure. In mathematics, it appears as equations, isomorphisms, and...

Error Bars

Error bars represent the range of uncertainty around a measured value. They show not just the estimate but how much that estimate could plausibly vary — encoding confidence,...

Frequentist Statistics

Frequentist statistics defines probability as the long-run frequency of events in repeated experiments. A coin has a 50% probability of heads not because of any belief about this...

Mean Median Mode

Mean, median, and mode are three ways to summarize the "center" of a dataset — and they tell different stories. The mean (average) includes every value, making it sensitive to...

Moderation

Moderation is the principle that the relationship between a variable and its outcome is often non-monotonic — more isn't always better, and the optimal point usually sits between...

Order of Approximation

Order of approximation is the practice of solving problems at the right level of precision for the decision at hand. A zeroth-order approximation ignores all nuance and gives a...

P-values

A p-value is the probability of observing data at least as extreme as the actual result, assuming the null hypothesis is true. It does not tell you the probability that your...

Permutations & Combinations

Permutations count the number of ways to arrange items when order matters. Combinations count selections when order doesn't. The distinction sounds academic but carries enormous...

Regression Analysis

Regression analysis models the relationship between variables — identifying how changes in one or more inputs predict changes in an output. Linear regression fits a straight line;...

Reliability of Case Evidence

Reliability of case evidence asks how much weight a single case — one customer story, one founder's success path, one market data point — should carry in forming beliefs....

Sampling

Sampling is drawing a subset from a larger population to make inferences about the whole. The power of sampling is efficiency — you don't need to survey every customer or test...

Statistical Significance

Statistical significance is the threshold at which an observed result is deemed unlikely enough under the null hypothesis to be considered "real" rather than random noise....

Stochastic Processes

A stochastic process is a system that evolves over time with inherent randomness — where the next state depends on the current state plus a random component. Stock prices,...

Surface Area

Surface area is the total exposed boundary between a system and its environment. In geometry, it determines how much of a solid contacts the outside world. In business, it...

Variability

Variability is the degree to which data points in a set differ from each other and from the central value. It's the spread, not the center. Two businesses can have identical...

Variance

Variance measures how far a set of outcomes spreads from the average. High variance means results swing widely; low variance means they cluster tight. The number itself is the...