Bayes Theorem. Susanna Kujanpää OUAS

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1 ayes Theorem Susanna Kujanpää OUS

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

3 ayes Theorem 2 Frequentist interpretation: it relates inverse representations of the probalilities concerning two events. This theorem has applications in a wide range of calculations insolving probabilities in statistics, science and engineering example marine biology and spam blockers for systems.

4 ayes Theorem If and are events, then where is conditional probability

5 ayes Theorem -> Then and n n n

6 ayes Theorem start n 1 n

7 ayes Theorem n n n n n

8 ayes Theorem EXMLE: Two computer Companies sell computer chips to a technology Company. Company sold 100 chips of which 5 were defective Company sold 300 chips of which 21 were defective What is the probability that a given defective chip came from Company?

9 ayes Theorem Solution: This information is given = the chip came from Company = the chip came from Company D = the chip is defective and D = 5 / 100 = 0.05 D = 21 / 300 = 0.07 and =?

10 ayes Theorem Now we have = 100 / 400 = 0.25 nd = 300 / 400 = 0.75 Then D D D D

11 ayes Theorem EXMLE: Lisa can decide to go study by a car, a bus or a train. ecause 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 20% and by train only 1%, but is more expensive than the bus.

12 ayes 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 each of the three possibilities. What is the teacher s estimate of the probability that Lisa drove to the school that day, given that she was late?

13 ayes Theorem Solution: This information is given = event Lisa comes by car = event Lisa comes by bus C = event Lisa comes by train D= event Lisa is late from school and = = C = 1/3 D = 0.5 D = 0.2 DC = 0.01

14 ayes Theorem We want to calculate and by ayes Theorem: = C C D D D D D

15 ayes Theorem In the special case of binary partition

16 ayes Theorem EXMLE Suppose it has been observed empirically that the word Congratulations occurs in 2 out of 10 spam s, but that Congratulations only occurs in 1 out of 1000 non-spam s. Suppose it has also been observed empirically that about 3 of 10 s are spam.

17 ayes Theorem This theorem helps to estimate the probability that a incoming is spam. Solution: = a new is spam = a new is contains Congratulations and = 3/10 = 0.3 = 7/10 = 0.7 = 2/10 = 0.2 = 1/1000 = 0.001

18 ayes Theorem Now by ayes Theorem, this is -> with high probability, such an is spam

19 ayes Theorem This example can be visualized with tree diagrams:

20 ayes Theorem nd the values are: 20% 6% 30% 80% 94% 70% 0,1% 30.1% 99.9% 69.9%

21 ayes Theorem EXMLE There are two box: The box 1 contains 2 red and 8 blue balls and the box 2 contains 7 red and 3 blue balls. Suppose Eric first randomly chooses one of two boxes 1 and 2, with equal probability, ½, of choosing each.

22 ayes Theorem Suppose Eric then randomly picks one ball out of the box he has chosen without telling you which box he had chosen, and shows you the ball he picked. Suppose you only see that the ball Eric picked is red. What is the probability that Eric chose box 1?

23 ayes Theorem Solution: = a event that box 1 was chosen = a event that a red ball was picked and = 1/2 = 0.5 = 1/2 = 0.5 = 2/10 = 0.2 = 7/10 = 0.7

24 ayes Theorem

25 EXERCISES: 1 teacher has a problem with the computer. The teacher knows that his information technology colleagues X could help him by the probability of 0.4 and Y by the probability of 0.5 and Z by the probability of 0.7. The teacher chooses randomly the colleague from whom he is going to ask for help. a Calculate the probability that the problem was solved. b The problem was solved. What is the probability that adviser was Z?

26 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. lso, 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.

27 3 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.

28 NSWERS: 1 a 0.53 b

29 Thank you!

This document contains Chapter 2: Statistics, Data Analysis, and Probability strand from the 2008 California High School Exit Examination (CAHSEE):

This document contains Chapter 2:, Data Analysis, and strand from the 28 California High School Exit Examination (CAHSEE): Mathematics Study Guide published by the California Department of Education. The