Formula and assumptions reference

C(n,k) = n! / [k! (n−k)!]

Unordered choice of k distinct items; order does not matter.

P(A∪B) = P(A)+P(B)−P(A∩B)

Addition rule; subtract overlap counted twice.

P(A|B) = P(A∩B) / P(B)

Requires P(B)>0; do not invert the conditioning.

P(H|+) = P(+|H)P(H) / P(+)

Bayes; P(+) must include all sources of a positive result.

P(X=k) = C(n,k)p^k(1−p)^(n−k)

Binomial; independent trials with fixed n and p.

E(X)=np; Var(X)=np(1−p)

Expectation and variance of a binomial count.

z = (x−μ) / σ

Normal standardisation, requiring σ>0.

SE(X̄) = σ/√n ≈ s/√n

First identity assumes iid finite-variance observations; s/√n estimates it.

CI: X̄ ± t_(0.975,n−1) × s/√n

Two-sided 95% t interval subject to sampling/model assumptions.

z = (X̄−μ0) / (σ/√n)

Idealised z test with known σ; p is not P(H0|data).

power = 1−β

Probability of rejecting H0 under a specified false-null alternative.

b = cov(X,Y) / s_X²; a = ȳ−bx̄

Simple OLS with an intercept; requires nonzero X variation.