1. COMP 170 L2 L16: Conditional Probability and Independence Page 1

2. COMP 170 L2 Outline Page 2 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

3. COMP 170 L2 Example: Roll Two Dice

4. COMP 170 L2 Event F and Change of Sample Spaces

5. COMP 170 L2 l F favors none of the three remaining possible outcomes l New probabilities n Preserve ratios n Sum to one

6. COMP 170 L2

10. Outline Page 10 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

11. COMP 170 L2

14. A Note

15. COMP 170 L2 Use of Conditional Probability Page 15

16. COMP 170 L2

17. Outline Page 17 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

18. COMP 170 L2 Motivating Example Page 18 l Learning that F occurred does not change probability of E l F is irrelevant to E l E is independent of F

19. COMP 170 L2 Definition Page 19

20. COMP 170 L2 Definition Page 20

21. COMP 170 L2 Outline Page 21 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

22. COMP 170 L2 Page 22

23. COMP 170 L2 Page 23

24. COMP 170 L2 Outline Page 24 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

25. COMP 170 L2 Page 25

26. COMP 170 L2 Page 26

27. COMP 170 L2 Page 27

28. COMP 170 L2 Page 28

29. COMP 170 L2 Outline Page 29 l Conditional probability n Motivating examples n Definition l Independence n Definition n Property n Examples l Independent trials processes

30. COMP 170 L2 Multiple Stage Process l Recall l Probability associated with a process with uncertain outcomes l Sample space: set of all possible outcomes l Event: subset of sample space l Probability distribution: weight for each element l One stage process l Flipping a coin once l Throwing two dice (all at once) l Multiple stage process l Flip coin multiple times l Hashing: Hash keys one at a time l Next: Discuss probabilities associated with multiple stage processes Page 30

31. COMP 170 L2 Independent Trial Process Outcome at stage i independent of outcomes at earlier stages Page 31

32. COMP 170 L2

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39. COMP 170 L2 Recap: 04-05-2010 l Probability starts with a process (test, experiment) whose outcome is uncertain l Sample space: the set of all possible outcomes n Might not be unique l Consider the probability of an event E n At the beginning, we have some assessment: P(E) n Then we get a new piece of information:  “Event F occurred” n Afterwards, we talk about the probability of E given F Page 39

40. COMP 170 L2 Recap: 04-05-2010 l If P(E|F) = P(E) n The information that “Event F occurred” does not change the probability of E n So, E is independent of F

41. COMP 170 L2 Recap: 06-05-2010 Outcome at stage i independent of outcomes at earlier stages