Bayes’ Rule Anchors: Olyvia Dean Viral Patel Eric Van Beek Group: Helium δ November 6, 2007

Joint Probability The probability of multiple events occurring: If the events are independent, their joint probability is the product of their individual probabilities:

Conditional Probability Used to find the probability of a certain event, A, given another event, B has occurred: If events A and B are dependent: If events A and B are independent:

Marginal Probability The probability that an event occurs, regardless of the outcome of any other event: Can be used if events are independent or dependent. If events are independent:

Bayes’ Rule Shows relationship between conditional and marginal probabilities Allows the probability of A to be adjusted given new information about past occurrences of B.

Drug Test Reliability BlueGoo requires drug tests for all of its employees. You test positive and are outraged! You decide to use Bayes’ rule to figure out the reliability of the drug test. Drug test is 98% accurate and it’s known that 1 in 1000 Americans use marijuana. Use Bayes’ Rule:

Drug Test Reliability (cont.) P(B|A) is the probability of testing positive if you use marijuana: 98%. P(A) is the probability that you use marijuana: 0.1% P(B) is the probability of the test giving a positive result; this is the sum of positive tests that were correct and positive tests that were incorrect: 98%*0.1% + 2%*99.9%. Only a 4.7% chance you actually use marijuana based on the test results and thus a 95.3% chance the results are wrong!

Big Ten Hopes If Michigan beats OSU what are the chances that Michigan will also win the Big Ten? 41 Big Ten titles Michigan vs. OSU record Michigan vs. OSU in Big Ten title year P(B|A) = 33.5 / 41 = P(A) = 41 / 103 = P(B) = 60 / 103 = Therefore:

Thank you for listening Bayes’ Rule