Conditional Probability Summary

Probability · Axiom Academy

SUMMARY Conditional Probability Key concepts, formulas, and problem-solving strategies Multiplication Rule: `P(A B) = P(A|B) P(B)` Bayes' Theorem: `P(A|B) = P(B) ` Independence: `P(A B) = P(A) P(B)` Law of Total Probability: Break down P(A) by partitioning the sample space Conditional Probability: Updates probability based on new information Independence: One event doesn't affect the other's probability Partitions: Mutually exclusive, exhaustive events that split the sample space Posterior Probability: Updated belief after observing evidence (Bayes) P(A|B) formula: You have P(A∩B) and P(B), need conditional probability Multiplication Rule: You have P(A|B) and P(B), need P(A∩B) Bayes' Theorem: You need to "reverse" conditional probabilities (know P(B|A), need P(A|B)) Law of Total Probability: Finding overall probability by breaking into cases/partitions Independence Check: Verify if P(A∩B) = P(A)·P(B) or if P(A|B) = P(A) Sequential Events: Use multiplication rule repeatedly for chain of conditions Example: Medical Testing with Bayes' Theorem Given: Disease rate P(D) = 0.01, Test accuracy P(+|D) = 0.95, False positive P(+|D c ) = 0.05 Find: If someone tests positive, what's P(D|+)? Strategy: Use Bayes' theorem with law of total probability for denominator Solution: `P(D|+) = P(+|D) P(D) + P(+|D^c) P(D^c) = 0.95 0.01 + 0.05 0.99 = 0.059 0.161` Interpretation: Only 16% chance of having disease despite positive test—base rate matters!

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