<!-- Generated from https://www.teclamairme.com/#group-T13 by readviews.py; never edited by hand. -->

# Glossary cards — Chapter ᚋ(M): The gallery of chance

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T13, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch13.md)

*Probability, information theory, statistics, and stochastic processes: a point with no width, Shannon’s bit, an average that tells the truth, Fisher’s one in twenty, and a weather chain that sits still. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Muin — `ᚋ` {#g-og-muin}

*Ogham letter*

Ogham letter name pulling three ways at once: ‘neck’, ‘ruse’, and ‘love’; the medieval glossators read it as the vine.

Muin is not a tree name, and the three kennings do not agree on which word it is: Old Irish has muin, ‘neck, upper part of the back’, muin, ‘wile, ruse’, and muin, ‘love, esteem’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tressam fedmae, “strongest in exertion”; árusc n-airlig, “proverb of slaughter”; and conar gotha, “path of the voice”—one for each sense. The arboreal tradition glosses finemhain, vine, with a kenning that reads the third sense: “highest of beauty is muin, that is, because it grows aloft.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### the probability of drawing her exactly — `P(X = ᚏ_{4.16})` {#g-probR}

*Integer or rational*

A point has no length, so under a continuous distribution such as the normal, the chance of one exact value is 0—a possible outcome with no chance, not an impossible one; Rose’s number crosses from the equation’s left side to its right, and measure theory (Chapter ᚓ(E)) is why.

Andrey Kolmogorov’s axioms of 1933 define a probability as a measure, and for any random variable with a density the probability of any single exact value is 0, since a point has zero length under the integral. This is the distinction between probability zero and impossibility: some value is always drawn, yet each particular one had chance 0. The number here is exactly 0, an integer, and it is the one place the left side’s undefinable number appears on the right, where measure theory (Chapter ᚓ(E)) gives it chance 0, though, like every value, it remains a possible draw.

Cites: [13.1](ch13-refs.md#r-13.1), [13.2](ch13-refs.md#r-13.2)

### the variance of a constant — `Var(c)` {#g-var0}

*Integer or rational*

A quantity that never moves has no spread: every deviation from its mean is 0, so its variance is 0.

Variance is the average squared distance from the mean. For a constant c the mean is c and every distance is 0, so Var(c) = 0; it is also the constant case of Var(aX + b) = a²Var(X). Exactly 0, an integer.

Cites: [13.25](ch13-refs.md#r-13.25), [13.3](ch13-refs.md#r-13.3)

### the covariance of independent variables — `Cov(X, Y), X ⟂ Y` {#g-cov0}

*Integer or rational*

Independent things do not move together, so their covariance is 0; the converse is famously false.

Covariance, E[(X − μₓ)(Y − μᵧ)], measures how two quantities move together. Independence forces E[XY] = E[X]E[Y] and so Cov = 0, but the converse fails: X and X² for X uniform on [−1, 1] have zero covariance and are dependent. Exactly 0 for independent variables with finite second moments, an integer.

Cites: [13.3](ch13-refs.md#r-13.3)

### the bell curve’s normalizer — `√(2π)` {#g-sqrt2pi}

*Proven transcendental*

The area under e^(−x²/2), the reason the normal distribution has a 2π in it; transcendental.

The integral of e^(−x²/2) over the whole line is √(2π). Laplace evaluated the probability integral in 1774 or 1782 (the sources differ); the familiar proof that squares the integral and changes to polar coordinates is credited to Poisson. The result supplies the normal density’s factor 1/√(2π), making its total probability 1. Transcendental, because π is: if √(2π) were algebraic, then 2π would be algebraic too.

Cites: [13.4](ch13-refs.md#r-13.4)

### the secretary-problem constant — `1/e` {#g-secretary}

*Proven transcendental*

With many candidates, pass over the first 37% and then take the first one better than all seen so far: both the skip fraction and the chance of success tend to 1/e; transcendental.

The secretary problem asks how to pick the best of n candidates interviewed in random order when each must be accepted or rejected on the spot; Martin Gardner popularized a related version in Scientific American in February 1960, and Dennis Lindley published the first solution of the classical secretary problem in 1961. Its optimal finite-n rule uses an integer cutoff near n/e and then takes the next candidate better than all seen so far. Both the cutoff fraction and the optimal success probability tend to 1/e ≈ 0.368 as n grows. Transcendental by Hermite’s 1873 theorem, since e is.

Cites: [13.28](ch13-refs.md#r-13.28), [13.5](ch13-refs.md#r-13.5)

### the probability of the impossible event — `P(∅)` {#g-pempty}

*Integer or rational*

0, by Kolmogorov’s axioms: the empty event contains no outcomes and therefore never happens.

Andrey Kolmogorov’s Grundbegriffe of 1933 made probability a measure on a σ-algebra with P(Ω) = 1, non-negativity, and countable additivity; P(∅) = 0 follows from additivity, since ∅ ∪ ∅ = ∅. It is the opposite of the numerator’s first zero: P(∅) = 0 because nothing is there, P(X = ᚏ) = 0 although something is. An integer.

Cites: [13.2](ch13-refs.md#r-13.2)

See also: [the probability of the sure event](ch13-cards.md#g-pomega), [a σ-algebra](ch20-cards.md#g-sigmaalg)

### the mean deviation from the mean — `E[X − E[X]]` {#g-meandev}

*Integer or rational*

0: deviations from the average average out, which is why variance squares them first.

Expectation is linear, so E[X − μ] = E[X] − μ = 0 for any X with finite mean, and the deviations of a sample from its arithmetic mean sum to zero for the same reason. The signed deviations cancel exactly; measures of spread therefore use absolute or squared deviations instead. An integer, 0.

Cites: [13.13](ch13-refs.md#r-13.13), [13.25](ch13-refs.md#r-13.25), [13.3](ch13-refs.md#r-13.3)

### the probability of the sure event — `P(Ω)` {#g-pomega}

*Integer or rational*

1: something happens. Kolmogorov’s second axiom, 1933.

The sample space Ω is the set of all outcomes and its probability is 1 by axiom; every probability lies between P(∅) = 0 and P(Ω) = 1. An integer.

Cites: [13.2](ch13-refs.md#r-13.2)

See also: [the probability of the impossible event](ch13-cards.md#g-pempty)

### the birthday problem — `23` {#g-birthday}

*Integer or rational*

In a room of 23 people the odds are better than even that two share a birthday; the smallest such crowd.

The birthday problem, first published by Richard von Mises in 1939 and generally attributed before that to Harold Davenport, who did not publish it at the time, asks how many people make a shared birthday more likely than not; the chance that n people all differ is 365!/(365^{n}(365 − n)!), which first drops below ½ at n = 23, where it is about 49.3%, so the shared-birthday chance is about 50.7%. The same calculation bounds the security of hash functions, the birthday attack. An integer.

Cites: [13.7](ch13-refs.md#r-13.7)

### the Monty Hall answer — `2/3` {#g-montyhall}

*Integer or rational*

Switching doors wins two times in three; the problem Steve Selvin posed in 1975 and Marilyn vos Savant made famous in 1990.

Three doors, one car; you pick one, the host, who knows where the car is, always opens a goat door you did not pick and always offers a switch. Switching wins with probability 2/3, because your first pick was wrong two times in three. Selvin published it in The American Statistician in 1975; vos Savant’s 1990 Parade column drew about 10,000 letters by the count reported at the time, most insisting she was wrong, including nearly 1,000 from readers with PhDs. A rational number.

Cites: [13.8](ch13-refs.md#r-13.8)

### the expected value of a die roll — `E[fair die]` {#g-diemean}

*Integer or rational*

7/2: the average of 1 through 6, a value the die never shows.

Expectation is the probability-weighted average of outcomes. For a fair die it is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5, even though no face shows 3.5: an expected value need not be an attainable outcome. A rational number.

Cites: [13.9](ch13-refs.md#r-13.9)

### the entropy of a certain event — `H(certain)` {#g-entropy0}

*Integer or rational*

Shannon entropy measures surprise, and an event with probability 1 carries none: H = −1·log 1 = 0.

Claude Shannon defined entropy in A Mathematical Theory of Communication, 1948, as the average of −log p over the outcomes of a source; an event with probability 1 contributes −1·log 1 = 0, so a certain event carries no information. Shannon noted that the logarithmic form is recognizable as entropy in statistical mechanics, linking the information measure to the older physical one. The value is exactly 0, an integer.

Cites: [13.10](ch13-refs.md#r-13.10)

### one bit — `1 bit` {#g-onebit}

*Integer or rational*

The entropy of a fair coin: the unit of information, Shannon 1948.

The bit, a contraction of binary digit suggested by John Tukey and adopted by Shannon in 1948, is the entropy of a fair coin: two outcomes, each with probability ½, give H = −2·(½ log₂ ½) = 1. It is the unit of binary information. The value is the integer 1, measured in bits.

Cites: [13.11](ch13-refs.md#r-13.11), [13.28](ch13-refs.md#r-13.28)

### the mutual information of independent variables — `I(X; Y), X ⟂ Y` {#g-mutinfo0}

*Integer or rational*

0 bits: knowing one tells you nothing about the other, Shannon 1948.

Mutual information I(X;Y) = H(X) + H(Y) − H(X, Y) measures in bits how much one variable tells you about another; in Shannon’s framework it is 0 exactly when X and Y are independent, unlike covariance, whose zero can hide dependence. It is the Kullback–Leibler divergence of the joint distribution from the product of the marginals. An integer, 0.

Cites: [13.12](ch13-refs.md#r-13.12)

### the divergence of a distribution from itself — `D_{KL}(P ‖ P)` {#g-kl0}

*Integer or rational*

0: the Kullback–Leibler divergence, the information lost by using one distribution in place of another, vanishes only when they are the same.

Solomon Kullback and Richard Leibler defined relative entropy D(P‖Q) = Σ p log(p/q) in 1951; it is non-negative by Gibbs’ inequality and 0 if and only if P = Q, though it is not a distance, being asymmetric. For P = Q the defining sum vanishes term by term. An integer, 0.

Cites: [13.12](ch13-refs.md#r-13.12)

### the sample mean is unbiased — `E[X̄] − μ` {#g-xbarunbiased}

*Integer or rational*

The average of a sample drawn from one distribution with a finite mean has the same expectation as one draw, so this is exactly 0.

For an i.i.d. sample X₁, …, Xₙ with finite mean μ, linearity of expectation gives E[X̄] = E[(X₁+⋯+Xₙ)/n] = μ, so the sample mean is an unbiased estimator of μ and E[X̄] − μ = 0 for every n. The result needs only integrability and an identically distributed sample, not normality or even independence; the displayed identity follows directly. Exactly 0, an integer.

Cites: [13.13](ch13-refs.md#r-13.13)

### OLS residuals sum to zero — `Σᵢ(yᵢ − ŷᵢ)` {#g-olsresid}

*Integer or rational*

With an intercept term, ordinary least squares chooses its line so that the residuals add up to exactly 0.

Ordinary least squares with an intercept minimizes Σ(yᵢ − a − bxᵢ)²; setting the derivative with respect to the intercept a to 0 gives Σ(yᵢ − ŷᵢ) = 0 directly, so the fitted residuals always sum to zero, whatever the data and whatever b turns out to be. The regression line through the data always passes through (x̄, ȳ) for the same reason. It fails only if the model omits the intercept. Exactly 0, an integer.

Cites: [13.14](ch13-refs.md#r-13.14), [13.25](ch13-refs.md#r-13.25)

See also: [least squares](ch10-cards.md#g-leastsq)

### the two-sided 95% critical value — `1.96` {#g-z196}

*Integer or rational*

The familiar rounded standard-normal value used to build a 95% confidence interval; approximately, not exactly, the 97.5th percentile.

1.96 approximates the 97.5th percentile of the standard normal distribution, the value z with Φ(z) = 0.975, used because a two-sided 95% interval excludes 2.5% in each tail. A confidence interval’s half-width is this critical value times a standard error, not the bare number 1.96 by itself, and 1.96 is only the conventional two-decimal approximation. The displayed decimal is itself a rational number.

Cites: [13.16](ch13-refs.md#r-13.16)

### Fisher’s significance level — `0.05` {#g-fisher005}

*Integer or rational*

One in twenty: Fisher’s convenient 1925 line for judging significance, a convention rather than a law of nature.

In Statistical Methods for Research Workers (1925) R. A. Fisher wrote that it is convenient to take a deviation of about twice the standard deviation, a probability of one in twenty, as the limit for judging a result significant. His wording presents one in twenty as a practical convention, not a universal theorem. A rational number.

Cites: [13.15](ch13-refs.md#r-13.15)

### a stationary Markov distribution — `‖πP−π‖_{1}` {#g-markovstationary}

*Integer or rational*

The distance between a Markov chain’s distribution and its next step; exactly 0 for P=[[1/2, 1/2], [1/4, 3/4]] and π=(1/3, 2/3), because π is stationary.

For the transition matrix P = [[1/2, 1/2], [1/4, 3/4]] and row distribution π = (1/3, 2/3), direct multiplication gives πP = π. Hence the ℓ¹ norm of πP−π is exactly 0. Markov introduced dependent chains in 1906; stationary distributions now organize random walks, mixing, and Markov-chain Monte Carlo. The concrete values make this a computation rather than a bare definition.

Cites: [13.17](ch13-refs.md#r-13.17)

See also: [a Markov chain](ch13-cards.md#g-markov)

### Pólya’s random-walk constant — `p₃` {#g-polya3}

*Status open*

About 0.3405: the chance that a drunk on a three-dimensional grid ever returns to where they started. On a line or a plane the return has probability 1.

George Pólya proved in 1921 that a simple random walk on the integer lattice returns to its origin with probability 1 in one and two dimensions, but not in three, where the return probability is p₃ = 1 − 1/u with u = (√6/(32π³))·Γ(1/24)Γ(5/24)Γ(7/24)Γ(11/24) ≈ 1.5164, so p₃ ≈ 0.340537 (Watson 1939, closed form Glasser and Zucker 1977). The current reference literature surveyed here gives exact forms for p₃ but no proof that it is irrational.

Cites: [13.18](ch13-refs.md#r-13.18), [13.24](ch13-refs.md#r-13.24), [13.28](ch13-refs.md#r-13.28)

### Bayes’ theorem — `P(A|B) = P(B|A)P(A)/P(B)` {#g-bayes}

*Function or operator*

How evidence updates belief; Bayes’s posthumous essay was presented by Richard Price in 1763, and Laplace developed an independent version in 1774.

Bayes’ theorem follows in one line from the definition of conditional probability P(A|B) = P(A ∩ B)/P(B); Thomas Bayes formulated a special case in an essay that Richard Price presented in 1763, after Bayes’s death, and Laplace independently developed and generalized the method in 1774. It is the engine of Bayesian inference and makes prior odds explicit when evidence updates a probability. A theorem, not a number.

Cites: [13.20](ch13-refs.md#r-13.20)

### the central limit theorem — `(X̄ − μ)√n/σ → N(0, 1)` {#g-clt}

*Function or operator*

Standardized averages of i.i.d. variables with finite variance tend to a bell curve: de Moivre’s early case, Laplace’s extension, and Pólya’s 1920 name.

In its familiar i.i.d. form, the central limit theorem says that standardized sums of variables with finite nonzero variance converge in distribution to the normal law. De Moivre found an early binomial case, Laplace extended it, later work supplied rigorous general versions, and George Pólya used the name central limit theorem in 1920. It is why √(2π) sits in the denominator: the bell curve is the limit all such standardized sums share. A theorem, not a number.

Cites: [13.21](ch13-refs.md#r-13.21)

See also: [the bell curve’s normalizer](ch13-cards.md#g-sqrt2pi)

### the law of large numbers — `X̄_{n} → μ` {#g-lln}

*Function or operator*

Averages settle down: Jacob Bernoulli’s theorem, published in Ars Conjectandi in 1713, is an early law of large numbers.

Jacob Bernoulli proved in Ars Conjectandi (1713) that an observed frequency concentrates near its probability as the number of trials grows. Later strong-law forms make the convergence almost sure. A theorem, not a number.

Cites: [13.22](ch13-refs.md#r-13.22)

### the standard normal density — `φ(x) = e^{−x²/2}/√(2π)` {#g-normal}

*Function or operator*

The bell curve, de Moivre 1733 and Gauss 1809: mean 0, variance 1, and area exactly 1 because of the √(2π) beneath it.

The normal distribution appeared as de Moivre’s limit of coin tosses in 1733, as Gauss’s law of errors in 1809, and as Laplace’s central limit in 1810; its density integrates to 1 precisely because ∫e^{−x²/2}dx = √(2π). About 68% of its mass lies within one standard deviation, 95% within two. A function, not a number.

Cites: [13.21](ch13-refs.md#r-13.21), [13.23](ch13-refs.md#r-13.23)

See also: [the central limit theorem](ch13-cards.md#g-clt), [the bell curve’s normalizer](ch13-cards.md#g-sqrt2pi)

### a Markov chain — `P(Xₙ₊₁ | X₀…Xₙ) = P(Xₙ₊₁ | Xₙ)` {#g-markov}

*Function or operator*

A process whose next step depends only on where it is now: Markov introduced these dependent chains in 1906.

A Markov chain is a sequence of random states in which the distribution of the next state depends on the present state alone, and not on the path that reached it. Andrei Markov’s 1906 paper introduced such dependent chains, extending the law of large numbers to variables that are not independent. Its transition matrix P collects the one-step probabilities, and a distribution π with πP = π is stationary, as the chapter’s two-state weather is. When the chain is finite, irreducible, and aperiodic—as that weather is, since every entry of its matrix is positive—the stationary distribution is unique and the chain walks toward it from any start at all, which is why the forecast forgets where it began. A process, not a number.

Cites: [13.17](ch13-refs.md#r-13.17), [13.26](ch13-refs.md#r-13.26)

See also: [the law of large numbers](ch13-cards.md#g-lln)