1. CSE 321 Discrete Structures Winter 2008 Lecture 19 Probability Theory TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAA A

2. Announcements Readings –Probability Theory 6.1, 6.2 (5.1, 5.2) Probability Theory 6.3 (New material!) Bayes’ Theorem 6.4 (5.3) Expectation –Advanced Counting Techniques – Ch 7. Not covered

3. Discrete Probability Experiment: Procedure that yields an outcome Sample space: Set of all possible outcomes Event: subset of the sample space S a sample space of equally likely outcomes, E an event, the probability of E, p(E) = |E|/|S|

4. Example: Dice

5. Example: Poker Probability of 4 of a kind

6. Combinations of Events E C is the complement of E P(E C ) = 1 – P(E) P(E 1  E 2 ) = P(E 1 ) + P(E 2 ) – P(E 1  E 2 )

7. Combinations of Events E C is the complement of E P(E C ) = 1 – P(E) P(E 1  E 2 ) = P(E 1 ) + P(E 2 ) – P(E 1  E 2 )

8. Probability Concepts Probability Distribution Conditional Probability Independence Bernoulli Trials / Binomial Distribution Random Variable

9. Discrete Probability Theory Set S Probability distribution p : S  [0,1] –For s  S, 0  p(s)  1 –  s  S p(s) = 1 Event E, E  S p(E) =  s  E p(s)

10. Examples

11. Conditional Probability Let E and F be events with p(F) > 0. The conditional probability of E given F, defined by p(E | F), is defined as:

12. Examples

13. Independence The events E and F are independent if and only if p(E  F) = p(E)p(F) E and F are independent if and only if p(E | F) = p(E)

14. Are these independent? Flip a coin three times –E: the first coin is a head –F: the second coin is a head Roll two dice –E: the sum of the two dice is 5 –F: the first die is a 1 Roll two dice –E: the sum of the two dice is 7 –F: the first die is a 1 Deal two five card poker hands –E: hand one has four of a kind –F: hand two has four of a kind

15. Bernoulli Trials and Binomial Distribution Bernoulli Trial –Success probability p, failure probability q The probability of exactly k successes in n independent Bernoulli trials is

16. Random Variables A random variable is a function from a sample space to the real numbers

17. Bayes’ Theorem Suppose that E and F are events from a sample space S such that p(E) > 0 and p(F) > 0. Then

18. False Positives, False Negatives Let D be the event that a person has the disease Let Y be the event that a person tests positive for the disease

19. Testing for disease Disease is very rare: p(D) = 1/100,000 Testing is accurate: False negative: 1% False positive: 0.5% Suppose you get a positive result, what do you conclude?

20. P(D|Y)

21. Spam Filtering From: Zambia Nation Farmers Union [znfukabwe@mail.zamtel.zm] Subject: Letter of assistance for school installation To: Richard Anderson Dear Richard, I hope you are fine, Iam through talking to local headmen about the possible assistance of school installation. the idea is and will be welcome. I trust that you will do your best as i await for more from you. Once again Thanking you very much Sebastian Mazuba.

22. Bayesian Spam filters Classification domain –Cost of false negative –Cost of false positive Criteria for spam –v1agra, ONE HUNDRED MILLION USD Basic question: given an email message, based on spam criteria, what is the probability it is spam

23. Email message with phrase “Account Review” 250 of 20000 messages known to be spam 5 of 10000 messages known not to be spam Assuming 50% of messages are spam, what is the probability that a message with “Account Review” is spam

24. Proving Bayes’ Theorem

25. Expectation The expected value of random variable X(s) on sample space S is:

26. Flip a coin until the first head Expected number of flips? Probability Space: Computing the expectation:

27. Linearity of Expectation E(X 1 + X 2 ) = E(X 1 ) + E(X 2 ) E(aX) = aE(X)

28. Hashing H: M  [0..n-1] If k elements have been hashed to random locations, what is the expected number of elements in bucket j? What is the expected number of collisions when hashing k elements to random locations?

29. Hashing analysis Sample space: [0..n-1]  [0..n-1] ...  [0..n-1] Random Variables X j = number of elements hashed to bucket j C = total number of collisions B ij = 1 if element i hashed to bucket j B ij = 0 if element i is not hashed to bucket j C ab = 1 if element a is hashed to the same bucket as element b C ab = 0 if element a is hashed to a different bucket than element b

30. Counting inversions Let p 1, p 2,..., p n be a permutation of 1... n p i, p j is an inversion if i p j 4, 2, 5, 1, 3 1, 6, 4, 3, 2, 5 7, 6, 5, 4, 3, 2, 1

31. Expected number of inversions for a random permutation

32. Insertion sort for i :=1 to n-1{ j := i; while (j > 0 and A[ j - 1 ] > A[ j ]){ swap(A[ j -1], A[ j ]); j := j – 1; } 42513

33. Expected number of swaps for Insertion Sort

34. Left to right maxima max_so_far := A[0]; for i := 1 to n-1 if (A[ i ] > max_so_far) max_so_far := A[ i ]; 5, 2, 9, 14, 11, 18, 7, 16, 1, 20, 3, 19, 10, 15, 4, 6, 17, 12, 8

35. What is the expected number of left-to- right maxima in a random permutation