Chris Morgan, MATH G160 csmorgan@purdue.edu January 20, 2012 Lecture 5 Chapter 4.5: Bayes’s Theorem Chris Morgan, MATH G160 csmorgan@purdue.edu January 20, 2012 Lecture 5

Bayes Theorem Extension of conditional probability: Usually I am given P(A|B) but what if I want to find the opposite?

Bayes Theorem Patients are tested for a certain disease and diagnosed as positive (having the disease) or negative (not having the disease), however these tests do not always detect the disease and sometimes give a false positive. - 1% of people will get a certain disease - 80% of those with the disease have “+” (positive) test results - 10% of those without the disease will have “+” (positive) test results Event D: a person has a certain disease Event +: a person tests positive Create a tree diagram representing this information.

Bayes Theorem Event D: a person has a certain disease Event +: a person tests positive P(+) = P(-) = P(+ | D) = P(+ | Dc) = P(D | +) =

P(Pass) = 0.9 P(Beach|Pass) = 0.8 Bayes Theorem After the first exam, a student will go to the beach (event B) depending on if they pass (event A): P(Pass) = 0.9 P(Beach|Pass) = 0.8 P(Beach|Not Pass) = 0.4 We can make a Tree Diagram to help us Visualize: Then, it’s easy to find: P(Beach) = P(Pass|Beach) =

Bayes Theorem Law of Total Probability: Bayes’ Rule: if A and AC form a partition of the sample space, then:

Bayes Theorem I’m trying to decide what type of question to put on your quiz. I’m leaning towards it being a conditional probability question with probability 75%, but I know that only 40% of you will get it correct. I know if I give you a question on Bayes Theorem, 80% of you will answer correctly. Draw the tree diagram to represent this data:

Bayes Theorem P(Bayes Question) = P(Correct | Cond Prob) = P(you solve the problem incorrect) = What is the probability I give you a Bayes Thm problem and you do it correctly? A={I give you a Bayes problem} B={you answer correctly} P(A and B) = P(B|A)*P(A) = 0.8 * 0.25 = 0.2

Bayes Theorem If a student solves the problem incorrectly, what is the probability that I gave you a Bayes Theorem problem? P(Bayes | Incorrect) =

Bayes Theorem (two event case)

Bayes Theorem (general case)

Example I Tomato seeds from supplier A have an 85% germination rate and those from supplier B have a 75% germination rate. A seed packaging company purchases 40% of their tomato seeds from supplier A and 60% from supplier B and mixes these seeds together. Draw the tree diagram to represent this data:

Example I (cont.) Find the probability that a seed selected at random from the mixed seeds will germinate. Given that a seed germinates, find the conditional probability that the seed was purchased from supplier A.

Example II A consulting firm submitted a bid for a large research project. The firm’s management initially felt they had a 50-50 chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for 75% of the successful bids and 40% of the unsuccessful bids the agency requested additional information. Draw the tree diagram to represent this data:

Example II (cont.) What is the prior probability of the bid being successful (that is, prior to the request for additional information? What is the conditional probability of a request for additional information given that the bid will ultimately be successful? Compute the posterior probability that the bid will be successful given a request for additional information?