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Conditional probability, independence, distribution types

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A Probability & Statistics set, untimed

Named traps

5 ways this goes wrong.

Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.

Key formulas

15
  • Classical probabilityP(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1P(certain)=1, P(impossible)=0
  • Combinations C(n,k)=n!/(k!(n−k)!)permutations A(n,k)=n!/(n−k)!
  • Addition ruleP(A∪B) = P(A) + P(B) − P(A∩B)
  • Complement: P(Ac) = 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)
  • Meanx̄ = (Σ xi)/nweighted/combined mean weights by group sizes
  • Variances2 = (1/n) Σ (xi − x̄)2standard deviation s = √s2range = xmax − xmin
  • Linear transformE(aX+b)=aE(X)+bD(aX+b)=a2 D(X)
  • Pooled variance of two groupsweight within-group variances by size AND add between-group (mean-shift) term
  • Normal PDFf(x) = 1/(σ√2π) · e−(x−μ2/(2σ2))curve symmetric about μtotal area = 1
  • Empirical rule≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
  • Standard normalZ = (x−μ)/σΦ(x)+Φ(−x)=1
  • Estimated count in an interval = P(interval) × population size
  • Least-squares slopeb̂ = (Σ xi yi − n x̄ ȳ)/(Σ xi2 − n x̄2)intercept â = ȳ − b̂ x̄
  • Normal approximation to binomial (De Moivre-Laplace)X~B(n,p) ≈ N(np, np(1−p))

Shaky underneath?

If the concept underneath still feels loose, patch it here before coming back to CSCA questions.

FoundationsStatistics and probability
Conditional probability, independence, distribution types — Probability & Statistics