Conditional Probability Notes from Stat 391 Conditional Events

Conditional Probability Notes from Stat 391

Conditional Events • We want to know P(A) for an experiment • Can event B influence P(A)? – Definitely! • Assume B is an experimental condition • We convey the dependence by P(A|B) – Means probability of A given B

Examples of Dependence • Rain Example – Probability that it will rain in the US in any city = P(rain) = 0. 3 – P(rain|Seattle) = 0. 5 – P(rain|Phoenix) = 0. 01 – P(rain|Seattle, summer) =. 25 • Image Example (contrived) – P(structure) = 0. 4 – P(structure|organized lines) = 0. 85

Properties of Conditional Probability • For disjoint sets, – P(A v C|B) = P(A v C, B)/P(B) = P(A, B) + P(C, B) / P(B) = P(A|B) + P(C|B) • Note: Under P(*|B), all outcomes that do not include B have a 0 probability • If A B, then – P(A|B) = P(A, B)/P(B) = P(A)/P(B) > P(A) – (i. e. , if A ↦ B, and B occurs, then the probability of A is increased)

More Properties of Conditional Probability • If B A, then – P(A|B) = P(A, B)/P(B) = P(B)/P(B) = 1 – If B ↦ A, then B occurring makes A certain • If B A = , then – P(A|B) = P(A, B)/P(B) = 0 • If A is conditioned on B, but A and B can never occur together, then clearly the probability of A when conditioned on B must be 0

Conditioning on Several Events • P(A|B, C) = P(A, B, C)/ P(B, C) = P(A, B|C)P(C)/P(B|C)P(C) = P(A, B|C)/P(B|C) – Same idea as before, but with C as context

Law of Total Probability • Law of Total Probability – Joint probability of A and B • P(A, B) = P(B)P(A|B) – Marginal probability of A • P(A) = P(A, B) + P(A, B^c) • P(A) = P(B)P(A|B) + P(B^c)P(A|B^c) • Example: Alice at a party (ex. 6, pg. 5)

Bayes’ Rule • From P(A, B) = P(B)P(A|B) = P(A)P(B|A)/P(B), we get Bayes’ Rule – P(A|B) = P(A)P(B|A)/P(B) • See court example (pg 6) for applicability

Independence • Independence means that knowing information about one event has no effect on the probability of the other • Formally, P(A, B) = P(A)P(B) – Or, P(A|B) = P(A) • Often expressed as A B to mean “A independent of B”

Example of Independence • Coin tosses are mutually independent – i. e. , knowing the outcome of one coin toss doesn’t change the probability of the outcome for another coin toss • Mutually independent events imply pairwise independence • Pair-wise independence doesn’t imply mutual independence