# Chapter ᚋ(M): The gallery of chance
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T13, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch13-cards.md) · [Its references](ch13-refs.md) · [Previous chapter](ch12.md) · [Next chapter](ch14.md)
*A footnote such as [r-13.3] is reference entry 13.3 and links to it. Section 13 of 28; chapters are named by their Ogham letter, never by number.*
ᚋ – [Muin](ch13-cards.md#g-og-muin): associated with the vine, Section 13
## Chapter ᚋ(M): The gallery of chance
> “The theory of probabilities is at bottom nothing but common sense reduced to calculus.”
> — Pierre-Simon Laplace · *A Philosophical Essay on Probabilities*, 1814 [r-13.27](ch13-refs.md#r-13.27)
Formula: ([P(X = ᚏ4.16)](ch13-cards.md#g-probR) + [P(∅)](ch13-cards.md#g-pempty) + [Var(c)](ch13-cards.md#g-var0) + [Cov(X, Y), X ⟂ Y](ch13-cards.md#g-cov0) + [E[X − E[X]]](ch13-cards.md#g-meandev))/(([√(2π)](ch13-cards.md#g-sqrt2pi) + [1/e](ch13-cards.md#g-secretary) + [2/π](ch04-cards.md#g-twoverpi)) {brace: constants} · ([P(Ω)](ch13-cards.md#g-pomega) + [23](ch13-cards.md#g-birthday) + [2/3](ch13-cards.md#g-montyhall) + [E[fair die]](ch13-cards.md#g-diemean)) {brace: famous answers} ) {brace: [probability](ch13.md#ch-T13-probability)} · ([H(certain)](ch13-cards.md#g-entropy0) + [I(X; Y), X ⟂ Y](ch13-cards.md#g-mutinfo0) + [DKL(P ‖ P)](ch13-cards.md#g-kl0))/([1 bit](ch13-cards.md#g-onebit) + [ln 2](ch04-cards.md#g-ln2)) {brace: [information theory](ch13.md#ch-T13-information-theory)} · ([E[X̄] − μ](ch13-cards.md#g-xbarunbiased) + [Σᵢ(yᵢ − ŷᵢ)](ch13-cards.md#g-olsresid))/([0.05](ch13-cards.md#g-fisher005) + [1.96](ch13-cards.md#g-z196)) {brace: [statistics](ch13.md#ch-T13-statistics)} · ([‖πP−π‖1](ch13-cards.md#g-markovstationary))/([p₃](ch13-cards.md#g-polya3)) {brace: [stochastic processes](ch13.md#ch-T13-stochastic-processes)}
#### Type of number
Proven transcendental · Status open ·
Key concepts: Probability, information theory, statistics, and stochastic processes. It includes symbols you may not have seen before, such as: [P](ch13-cards.md#g-pempty), [Var](ch13-cards.md#g-var0), [H](ch13-cards.md#g-entropy0), and [DKL](ch13-cards.md#g-kl0). Zero here is certainty and indifference: the chance of drawing Rose’s exact number, the surprise of a sure thing, and a chain whose odds have stopped changing, though its weather has not.
The gallery of chance is arranged like a weather station. Its instruments record what is happening, while the forecast reaches toward what may happen next. I would like a guarantee of sunshine for the road ahead, but we will have to settle for something more interesting: a way to describe uncertainty carefully. Probability, information, statistics, and stochastic processes each have a part in that description. Rose joins us again at the first display: her number, ᚏ4.16, is the exact value we will try to draw, and the chance of drawing it turns out to be zero.
Field notes
### [probability](ch13.md#ch-T13-probability) {#ch-T13-probability}
Probability assigns each event a number between 0 and 1. A zero here can mean two very different things: an outcome too thin to carry any weight, or no outcome at all.
#### Numerator
[P(X = ᚏ4.16)](ch13-cards.md#g-probR) = 0; [Rose’s number](ch01-cards.md#g-rose), ᚏ4.16, returns from the left side for the first time since the gateway, as a possible outcome with zero chance.
Let X have [a normal distribution](ch13-cards.md#g-normal), whose bell-shaped density is positive across the [real line](ch01-cards.md#g-reals). The chance of landing in a range of values is the area under the curve above that range; a single point has no width and so no area, and every exact value has probability 0. Yet the draw has an outcome: probability zero is not the same as impossibility. We need no decimal description of her number, because this holds for every fixed real value. [r-13.1](ch13-refs.md#r-13.1)
[P(∅)](ch13-cards.md#g-pempty) = 0; a zero for the opposite reason, placed beside hers to show the difference.
The empty event contains no possible outcomes at all, and the rules of probability assign it zero. [r-13.2](ch13-refs.md#r-13.2)
[Var(c)](ch13-cards.md#g-var0) = 0; the simplest fact about spread.
A constant never varies, so it has no spread. [r-13.3](ch13-refs.md#r-13.3) [r-13.25](ch13-refs.md#r-13.25)
[Cov(X, Y), X ⟂ Y](ch13-cards.md#g-cov0) = 0; a fundamental property of independent quantities, with a famous trap in its converse.
Covariance measures whether two quantities tend to rise and fall together. When X and Y are independent, and their spreads are finite, their covariance is zero. The reverse fails: let X be equally likely to land anywhere between −1 and 1, and let Y = X². Symmetry gives Cov(X, Y) = 0, yet knowing X determines Y exactly. Covariance sees only straight-line association. [r-13.3](ch13-refs.md#r-13.3)
[E[X − E[X]]](ch13-cards.md#g-meandev) = 0; the balance point that defines an average.
Subtracting the mean leaves a quantity whose mean is zero, whenever X has a finite mean: the ups and downs cancel exactly. [r-13.13](ch13-refs.md#r-13.13) [r-13.25](ch13-refs.md#r-13.25) That is why variance squares them before averaging. [r-13.3](ch13-refs.md#r-13.3)
#### Denominator
Two sums, multiplied: a bell curve, choosing the best candidate, and a needle crossing a line; then the whole sample space, birthdays, doors, and dice. Each example has its own model.
[√(2π)](ch13-cards.md#g-sqrt2pi) ≈ 2.5066; the constant behind the bell curve.
It is the area under the curve e−x²/2 across the whole [real line](ch01-cards.md#g-reals); dividing the curve by it makes the bell’s total area, its total probability, exactly 1. [r-13.4](ch13-refs.md#r-13.4) [r-13.23](ch13-refs.md#r-13.23) The [central limit theorem](ch13-cards.md#g-clt) is why this curve turns up so often: add up many independent quantities with finite variance, shift and rescale the total, and its distribution approaches the bell. [r-13.21](ch13-refs.md#r-13.21)
[1/e](ch13-cards.md#g-secretary) ≈ 0.3679; the answer to the classic secretary problem.
Candidates arrive in random order, and each must be accepted or turned away on the spot, judged only against those already seen. The best plan is to let an opening share of them go by, then take the first one better than everyone so far. As the number of candidates grows, that opening share and the chance of ending up with the very best both approach 1/e, about 37%. [r-13.5](ch13-refs.md#r-13.5)
[2/π](ch04-cards.md#g-twoverpi) ≈ 0.6366; Buffon’s needle, where π turns up in a game of chance.
Drop a needle as long as the spacing between parallel lines, with position and angle chosen uniformly; it crosses a line with probability [2/π](ch04-cards.md#g-twoverpi). [r-13.6](ch13-refs.md#r-13.6)
[P(Ω)](ch13-cards.md#g-pomega) = 1; the rule that something must happen.
The whole sample space contains every outcome, so its probability is 1.
[23](ch13-cards.md#g-birthday) = 23; the birthday paradox, famous for sounding too small.
With 23 people, a shared birthday is more likely than not, in the model where birthdays are independent and uniform over 365 days. [r-13.7](ch13-refs.md#r-13.7)
[2/3](ch13-cards.md#g-montyhall) = 2/3; the Monty Hall problem, famous for starting arguments.
Switching wins two times in three, in the standard setup: the host knows where the prize is, always opens a losing door you did not choose, and always offers the switch. [r-13.8](ch13-refs.md#r-13.8)
[E[fair die]](ch13-cards.md#g-diemean) = 7/2; an average no roll can show.
The mean of a fair six-sided die is (1 + 2 + ⋯ + 6)/6 = [3½](ch13-cards.md#g-diemean), a value the die never displays. [r-13.9](ch13-refs.md#r-13.9) Roll many times and the running average settles toward it anyway, which is the [law of large numbers](ch13-cards.md#g-lln). [r-13.22](ch13-refs.md#r-13.22)
The first fraction is 0/(positive product): ([√(2π)](ch13-cards.md#g-sqrt2pi) + 1/e + [2/π](ch04-cards.md#g-twoverpi))(1 + [23](ch13-cards.md#g-birthday) + [2/3](ch13-cards.md#g-montyhall) + 7/2); both sums are positive, so their product is too.
**Fine print.** Here Ω names the sample space, not [Chaitin’s halting probability](ch05-cards.md#g-omega). Each denominator example lives in its own model. A particular sample’s deviations from the population mean need not sum to zero; [E[X − E[X]]](ch13-cards.md#g-meandev) = 0 is a statement about expectation.
### [information theory](ch13.md#ch-T13-information-theory) {#ch-T13-information-theory}
Shannon’s theory measures information as removed uncertainty. Its zeros are three different vanishings: no uncertainty, no dependence, and no discrepancy.
#### Numerator
[H(certain)](ch13-cards.md#g-entropy0) = 0; the entropy of a sure thing, Shannon’s starting point.
Entropy averages the surprise of a source’s outcomes, and an outcome with probability p carries surprise −log p: the rarer it is, the bigger the surprise. A sure outcome has probability 1 and surprise −log 1 = 0. A message that can only say one thing removes no uncertainty, like a forecast that promises only that tomorrow will have weather. [r-13.10](ch13-refs.md#r-13.10)
[I(X; Y), X ⟂ Y](ch13-cards.md#g-mutinfo0) = 0; chosen because it repairs the covariance trap above.
Mutual information measures what learning one variable tells us about the other, and for independent variables that is nothing. Unlike covariance, zero mutual information characterizes independence, so it does not miss a relationship such as Y = X² just because it is nonlinear. [r-13.12](ch13-refs.md#r-13.12)
[DKL(P ‖ P)](ch13-cards.md#g-kl0) = 0; a distribution compared with itself.
The Kullback–Leibler divergence measures how one probability distribution differs from another, and a distribution does not differ from itself. [r-13.12](ch13-refs.md#r-13.12)
#### Denominator
[1 bit](ch13-cards.md#g-onebit) = 1; the unit of information: a fair coin’s entropy in base-2 logarithms.
A fair coin has two equally likely outcomes, and −log₂ ½ = 1. [r-13.11](ch13-refs.md#r-13.11)
[ln 2](ch04-cards.md#g-ln2) ≈ 0.6931; the same coin measured with natural logarithms.
The same entropy, −ln ½ = ln 2, now counted in nats: the nat is the unit of information that natural logarithms give, as the bit is for base-2 logarithms. [r-13.29](ch13-refs.md#r-13.29)
The second fraction is 0/(1 + ln 2): 1 + ln 2 is positive.
**Fine print.** The denominator adds the displayed numerical readings, 1 and ln 2, after recording their units; it does not claim that bits and nats can be added without conversion. “Divergence” is deliberate: the Kullback–Leibler measure is generally asymmetric and is not an ordinary distance.
### [statistics](ch13.md#ch-T13-statistics) {#ch-T13-statistics}
Statistics turns the gallery’s observations into estimates. Its zeros say that a method is balanced. Its denominator holds two familiar conventions of statistical reporting: a one-in-twenty threshold and the bell-curve cutoff associated with it.
#### Numerator
[E[X̄] − μ](ch13-cards.md#g-xbarunbiased) = 0; the sample mean, the most basic unbiased estimate.
Let X̄ be the average of several observations that each have the same finite mean μ. The expected value of an average is the average of the expected values, so E[X̄] = μ; the observations need not even be independent. *Unbiased* means right on average over repeated samples, not that every sample mean equals μ. [r-13.13](ch13-refs.md#r-13.13)
[Σᵢ(yᵢ − ŷᵢ)](ch13-cards.md#g-olsresid) = 0; a basic property of [least-squares fitting](ch10-cards.md#g-leastsq).
A *residual* is an observed value minus the fitted value. In ordinary [least squares](ch10-cards.md#g-leastsq) with a freely fitted intercept, the best fit has residuals summing to zero; otherwise, shifting the whole line slightly up or down would reduce the sum of squared residuals. Even a perfectly balanced fit need not predict the next observation exactly. [r-13.14](ch13-refs.md#r-13.14) [r-13.25](ch13-refs.md#r-13.25)
#### Denominator
[0.05](ch13-cards.md#g-fisher005) = 0.05; Fisher’s one in twenty, the conventional significance threshold.
It is a recognizable choice from statistical practice, not a universal rule for deciding what data mean. [r-13.15](ch13-refs.md#r-13.15)
[1.96](ch13-cards.md#g-z196) = 1.96; the cutoff behind familiar 95% intervals.
It is the rounded standard-normal value used in two-sided 95% intervals: 95% of the bell’s area lies within about 1.96 standard deviations of the middle, and the other 5% in the two tails beyond. [r-13.16](ch13-refs.md#r-13.16) So it is the 0.05 above seen from the other side; Fisher himself wrote that the value for which P = .05, or 1 in 20, “is 1.96 or nearly 2.” [r-13.15](ch13-refs.md#r-13.15)
The third fraction is 0/2.01: 0.05 + 1.96 = 2.01.
**Fine print.** A probability and a quantile are different kinds of number; the denominator adds their displayed readings, as the information brace adds a bit and a nat. The intercept condition is essential to the residual sum, and numerical software may leave a small rounding residual. The displayed 1.96 is itself a rounded decimal of the underlying normal quantile; the sum of the displayed values is exactly 2.01.
### [stochastic processes](ch13.md#ch-T13-stochastic-processes) {#ch-T13-stochastic-processes}
Randomness unfolding over time. In [a Markov model](ch13-cards.md#g-markov), the next state depends only on the current one; earlier history adds no information. Matrices from Chapter ᚈ(T) now describe uncertainty moving through time.
#### Numerator
[‖πP−π‖1](ch13-cards.md#g-markovstationary) = 0; a weather forecast built to sit still.
Take states rain and shine with transition matrix P = [[½, ½], [¼, ¾]]: rain is followed by rain or shine equally often, and shine is followed by rain one time in four. Start with rain probability 1/3 and shine 2/3. Tomorrow’s rain probability is (1/3)(1/2) + (2/3)(1/4) = 1/3 again, so π = (1/3, 2/3) is *stationary*: πP = π, and the norm of the difference is 0. The distribution stays fixed while the weather itself keeps changing. [r-13.17](ch13-refs.md#r-13.17)
#### Denominator
[p₃](ch13-cards.md#g-polya3) ≈ 0.3405; Pólya’s famous discovery that a wanderer in three dimensions may never come home.
Picture a wanderer on an endless three-dimensional grid of points who, at every step, moves to one of the six neighboring points, each equally likely: forward or back, left or right, up or down. p₃ is the chance that the wanderer ever comes back to the start. Pólya showed that on a line or a flat grid the return is certain, with probability 1; in three dimensions the chance is strictly between 0 and 1. [r-13.18](ch13-refs.md#r-13.18)
The fourth and final fraction is 0/[p₃](ch13-cards.md#g-polya3): [p₃](ch13-cards.md#g-polya3) is positive, even though return is not certain.
**Fine print.** The ‖·‖₁ norm adds the absolute differences of the coordinates. The π here is the conventional letter for a stationary distribution, not the circle constant earlier in the chapter.
What this chapter adds to the formula: 0/(positive product) · 0/(1 + ln 2) · 0/2.01 · 0/[p₃](ch13-cards.md#g-polya3) = 0
These zeros have described a point with no probability mass, no uncertainty, balanced deviations, and an unchanged distribution. The gallery has also brought our tools together: geometry supplies area, matrices evolve probabilities, and calculus finds a fitted line.
We can leave the forecast running. The weather kept changing while its distribution stayed the same; the next room asks what else a transformation can preserve. Enough worrying about the weather. Let’s have some fun in the next exhibit—the hall of mirrors.
Watch and read
- [Bayes theorem, the geometry of changing beliefs](https://www.youtube.com/watch?v=HZGCoVF3YvM) 3Blue1Brown; probability as area, which is the picture that makes a single point’s probability zero obvious
- [Pi and Buffon’s Matches - Numberphile](https://www.youtube.com/watch?v=sJVivjuMfWA) Tony Padilla drops matches to measure π, the 2/π in the denominator, live
- [Probability axioms](https://en.wikipedia.org/wiki/Probability_axioms) Wikipedia
- [Entropy (information theory)](https://en.wikipedia.org/wiki/Entropy_(information_theory)) Wikipedia