An architectural exploration of probability theory — where each concept is a precisely drafted technical diagram rendered in blueprint-blue on white.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The sample space Ω contains all possible outcomes. Events are subsets of this space, and their probabilities follow the axioms of measure.
P(X = k) = (n choose k) · p^k · (1−p)^(n−k)
Discrete probability distributions map each outcome to its likelihood. The binomial distribution counts successes in independent trials — each bar represents the probability of exactly k successes.
P(H|E) = P(E|H) · P(H) / P(E)
Bayes' theorem inverts conditional probabilities — updating our belief in hypothesis H given evidence E. The tree diagram traces all paths through which evidence can arise.
X̄ₙ → N(μ, σ²/n) as n → ∞
As sample size increases, the distribution of sample means converges to a normal distribution regardless of the population's shape. The bell curve emerges from chaos — order from randomness.
P(Xₙ₊₁ = j | Xₙ = i) = pᵢⱼ
A Markov chain is a stochastic process where future states depend only on the present — the memoryless property. State transitions are governed by the probability matrix P.