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Probability Glossary

The core vocabulary — sets and events, the axioms, conditioning, random variables, and the common distributions — each with a small picture.

Sample Spaces & Events

Ω Sample space & outcomes
The set of all possible outcomes of an experiment. A single outcome ω is one point of Ω — one way the experiment can turn out. Everything else is built from this set.
A die roll: Ω = {1,2,3,4,5,6}. A coin: Ω = {H, T}.
A Event
Any subset of the sample space — a collection of outcomes we care about. The event "occurs" when the realised outcome ω falls inside A.
"Roll is even" = {2, 4, 6} ⊆ Ω.
Aᶜ Complement
The event that A does not occur — every outcome in Ω that is outside A. Its probability is whatever is left over after A.
A∪B Union
The event that at least one of A, B occurs — every outcome lying in A, in B, or in both. Read as "A or B".
A∩B Intersection
The event that both A and B occur — outcomes in the overlapping lens. Read as "A and B".
A∩B = ∅ Mutually exclusive
Two events that cannot both happen — their circles never touch. Then P(A∪B) = P(A) + P(B), with no overlap to subtract.
Disjoint ≠ independent. Disjoint events are in fact strongly dependent: if one occurs, the other cannot.

Axioms & Rules

P Probability measure
A rule assigning each event a number in [0, 1] — think of it as area, where the whole sample space has area 1. Probabilities of disjoint events add.
Kolmogorov's three axioms: non-negativity, P(Ω) = 1, and countable additivity.
A∪B Addition rule (inclusion–exclusion)
To find P(A∪B) you add the two areas, but the overlap got counted twice — so subtract it once. For disjoint events the overlap is zero.
(A∪B)ᶜ De Morgan's laws
"Not (A or B)" is the same as "not A and not B" — the shaded region outside both circles. Complementing flips union ↔ intersection.
Dually, (A∩B)ᶜ = Aᶜ ∪ Bᶜ.

Conditioning & Independence

P(A|B) Conditional probability
The probability of A once we know B occurred. B becomes the new sample space; we ask what fraction of B also lies in A. In general P(A|B) ≠ P(B|A).
A ⫫ B Independence
Knowing B tells you nothing about A: P(A|B) = P(A). Equivalently the joint probability factorises into the product — the overlap area equals width × height.
Independent is not the same as disjoint, and pairwise independence does not imply mutual independence.
Bayes Bayes' rule & total probability
A partition B₁,…,Bₙ slices Ω into disjoint pieces. Total probability reassembles P(A) by summing A's share of each slice; Bayes then inverts a conditional to update beliefs from evidence.
Posterior ∝ likelihood × prior.

Random Variables & Moments

X Random variable
A function that attaches a number to every outcome — turning qualitative outcomes into something we can average and plot. Discrete takes isolated values; continuous takes a whole range.
p(x) Probability mass function (PMF)
For a discrete variable: the probability of each individual value, P(X = x). The spikes are genuine probabilities and must sum to 1.
f(x) Probability density function (PDF)
For a continuous variable: probability is area under the curve. The height f(x) is not a probability — only the area over an interval is. Total area = 1.
F(x) Cumulative distribution (CDF)
The probability of landing at or below x — it accumulates the mass/area from the left, rising monotonically from 0 to 1. Works for discrete and continuous alike.
E[X] Expectation (mean)
The probability-weighted average value — the balance point of the distribution. Place the density on a seesaw and μ is where it tips level.
Var(X) Variance & standard deviation
The expected squared distance from the mean — how spread out the distribution is. The standard deviation σ = √Var is the spread in the variable's own units.

Common Distributions

Bin(n,p) Binomial
The count of successes in n independent yes/no trials, each with success probability p. A single trial (n = 1) is the Bernoulli distribution. Mean np, variance np(1−p).
Pois(λ) Poisson
The count of rare events in a fixed interval when they occur independently at average rate λ. Mean and variance are both equal to λ.
N(μ,σ²) Normal (Gaussian)
The symmetric bell curve centred at μ with spread σ. By the central limit theorem sums and averages of many independent effects drift toward it — which is why it appears everywhere.
≈ 68% of mass within ±1σ, ≈ 95% within ±2σ.
Exp(λ) Exponential
The waiting time until the next Poisson event. Memoryless: the time already waited tells you nothing about the time remaining. Mean 1/λ.

Explore any of these interactively in the Distribution Calculator — sliders for the parameters, live PMF/PDF and CDF, and point queries.