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

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

4. 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.

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

6. 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) = 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) =

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

8. 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:

9. 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

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

11. Bayes Theorem (two event case)

12. Bayes Theorem (general case)

13. 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:

14. 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.

15. Example II
A consulting firm submitted a bid for a large research project. The firm’s management initially felt they had a 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:

16. 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?