1. Conditional Probability Notes from Stat 391

2. 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

3. 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

4. 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)

5. 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/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

6. 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

7. 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)

8. Bayes’ Rule From P(A, B) = P(B)P(A|BP(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

9. 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”

10. 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 pair- wise independence Pair-wise independence doesn’t imply mutual independence