ConceptProbability & Statistics

Key formulas

Classical probability: P(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1; P(certain)=1, P(impossible)=0
Combinations C(n,k)=n!/(k!(n−k)!); permutations A(n,k)=n!/(n−k)!
Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)
Complement: P(A^c) = 1 − P(A); at-least-one: P(≥1) = 1 − P(none)
Conditional: P(A|B) = P(A∩B)/P(B); Multiplication: P(A∩B)=P(A)P(B|A); Independence: P(A∩B)=P(A)P(B)
Mean: x̄ = (Σ x_i)/n; weighted/combined mean weights by group sizes
Variance: s² = (1/n) Σ (x_i − x̄)²; standard deviation s = √s²; range = x_max − x_min
Linear transform: E(aX+b)=aE(X)+b; D(aX+b)=a² D(X)
Pooled variance of two groups: weight within-group variances by size AND add between-group (mean-shift) term
Normal PDF: f(x) = 1/(σ√2π) · e^−(x−μ²/(2σ²)); curve symmetric about μ; total area = 1
Empirical rule: ≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
Standard normal: Z = (x−μ)/σ; Φ(x)+Φ(−x)=1
Estimated count in an interval = P(interval) × population size
Least-squares slope: b̂ = (Σ x_i y_i − n x̄ ȳ)/(Σ x_i² − n x̄²); intercept â = ȳ − b̂ x̄
Normal approximation to binomial (De Moivre-Laplace): X~B(n,p) ≈ N(np, np(1−p))

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FoundationsStatistics and probability Precalculus bridge

Probability from real data charts (air-quality index, frequency bar/line charts) including joint multi-day conditions

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