1. Bayes’ Theorem Susanna Kujanpää OUAS 6.12.2012

2. Bayes’ Theorem This is a theorem with two distinct interpretations. 1) Bayesian interpretation: it shows how a subjective degree of belief should rationally change to account for evidence

3. Bayes’ Theorem 2) Frequentist interpretation: it relates inverse representations of the probalilities concerning two events.

4. Bayes’ Theorem If A and B are events, then where P(A|B) is conditional probability and

5. Bayes’ Theorem In the special case of binary partition

6. Bayes’ Theorem EXAMPLE: Lisa can decide to go study by car or bus. Because of high traffic, if she decides to go by car, there is a 50% chance she will be late. If she goes by bus, the probability of being late is only 20%.

7. Bayes’ Theorem Suppose that Lisa is late one day, and her teacher wishes to estimate the probability that she drove to school that one day by car. Since she doesn’t know which mode of transportation Lisa usually uses, she gives a prior probability of 1/3 to both possibilities. What is the teacher’s estimate of the probability that Lisa drove to the school ?

8. Bayes’ Theorem Solution: This information is given A = event Lisa comes by a car = Lisa comes by a bus B= event Lisa is late from school and P(A) = P( ) = 1/3 P(B|A) = 0.5 P(B| ) = 0.2

9. Bayes’ Theorem We want to calculate P (A|B) and by Bayes Theorem, this is =

10. Bayes’ Theorem This example can be visualized with tree diagrams:

11. Bayes’ Theorem And the values are:

12. Bayes’ Theorem EXAMPLE Lisa is graduating at outdoor ceremony tomorrow. In recent years, it has rained only one week each year. Now, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. What is the probability that it will rain on graduating day ?

13. Bayes’ Theorem Solution: A = It rains on graduating day = It doesn’t rain on graduating day B= weatherman predicts rain and P(A) = 7/365 = 0.019 P( ) = 358/365 = 0.981 P(B|A) = 0.9 P(B| ) = 0.1

14. Bayes’ Theorem Now by Bayes Theorem, this is

15. EXERCISES: 1) Suppose that Mike can decide to go work by bus, car or train. If he decides to go by car, there is 45% chance he will be late. If he goes by bus, there is 25% chance of being late. The train is almost never late, with probability of 2%, but it is more expensive than the bus. Suppose that Mike is late one day and his friend wishes to estimate the probability that he drove to work that day by car (he gives the probability of 1/3 to each possibilities). What is the friend’s estimate of the probability that Mike drove to work ?

16. 2) In the village, 51% of the adults are males. One of the adult is randomly selected for a survey involving credit card usage. Later was noticed that the selected survey subject was using headache medicin. Also, 9.5% of males use medicin and 1.7% of females. Use this additional information to find the probability that the selected subject is male.

17. 3) A first company makes 80% of the products, the second makes 15% of them and the third makes the other 5%. The first company have a 4% rate of defects, the second have a 6% rate of defects and the third have a 9%. If a randomly selected product is then tested and is found to be defective, find the probability that it was made by the first company.

18. ANSWERS: 1) 0.625 2) 0.853 3) 0.703