1. Applying Probability Define problem of interest – in terms of “random variables” and/or “composite events” Use real world knowledge, symmetry – to associate probs in [0,1] with ‘elementary events’ – all probs are conditional on real world knowledge Use consistent prob rules – To associate probs with rand vars/ comp events – Multiplication and Addition Rules 1ST2004 Week 7

2. Probability Prob Rules Week 7 – Basic in text, Ch2.2 – Conditional Prob Bayes Rule in Ch 6 – Fuller treatment in Ch 7 8 Discrete Prob Dist Week 8 – Ch 4 – see lab on Queuing – Ch 9 Continuous Prob DistWeek 9 – Ch 5Normal dist – Ch 10 ST2004 Week 72 We give more emphasis to ‘event identities’. Book in Ch 7 uses more math shortcuts (binomial coeffs) and  notation than we will use. Best immediate preparation is Q1-12 in Ch 1. Formulate and approach via EXCEL before attempting probability solution.

3. ST2004 2010 Week 6 Google’s PageRank Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by Google are rescaled logarithmically). Page C has a higher PageRank than Page E, even though it has fewer links to it; the link it has is of a much higher value. A web surfer who chooses a random link on every page (but with 15% likelihood jumps to a random page on the whole web) is going to be on Page E for 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages A, B, or C, and all other pages would have PageRank zero. Page A is assumed to link to all pages in the web, because it has no outgoing links. Pr( Next goes to page C, given now at A) 3

4. Cards and Queens Simulation Probability 4ST2004 Week 7

5. Mini-League Probs 5ST2004 Week 7

6. Problems Dice: Seek prob dist of M 2,S 2,M 3,S 3,M k,S k – LaterE(S 2 )Var(S 2 ) etc Mini-league: Seek prob dist of (N A, N B, N C ) when – Pr( A beats B)=2 Pr(B beats A)  Pr( A beats B)=? – Pr( A beats B)=p AB ; similarly p BC, p AC – LaterE(N A ),Var(N A ) and E(N A |N C =0),Var(N A |N C =0) ST2004 Week 76

7. ST2004 2010 Week 6 Events, Random Vars, Sample Space and Probability Rules Event A Simplest Random Variable Values of A are TRUE/FALSE Random Variable Y Values of Y are y 1, y 2..y k (sample space; exhaustive list) Events such as (Y= y) 7

8. ST2004 2010 Week 6 Event Algebra Event Identities Re-express compound events in and/or combinations of elementary events Coin (H orT)  Experiment Happened Cards Ace  (A ♠ orA ♥ or A ♣ or A ♦ ) Redand (NOT ♦ )  (2 ♥ or.. or A ♥ ) 8

9. ST2004 2010 Week 6 Event Identities Re-express in terms of and/or combs of (..) (elementary events and/or simple compound events). Often there is more than one way. “A out-right winner of league”. Use as elementary events Outcomes of games A/B, etc, and as relatively simple compound events, the scores N A, etc “At least one Queen in two cards” “Max of 3 dice is 3” and “Max of 3 dice is  3” “Sun of 3 is 4” 9

10. Event Identities 10ST2004 Week 7

11. Event Identities: Coin Toss ST2004 Week 711

12. Event Identities: Password ST2004 Week 712

13. Event Identities 13ST2004 Week 7

14. Event Identity etc 14ST2004 Week 7

15. Event Identity etc 15ST2004 Week 7

16. Event Identities for Random Vars 16ST2004 Week 7

17. Probability Rules Addition Rule 17ST2004 Week 7 Plus real world knowledge

18. Coins/Dice/Cards 18ST2004 Week 7

19. Applying Prob Rules Generalisation of Addition Rule 19ST2004 Week 7

20. Event Identities: Password ST2004 Week 720 Elementary events and associated probs Pr(Dup) via addition rules

21. Conditional Probability 21ST2004 Week 7

22. Conditional Simulation Sequence of simulations – simulation rules at each stage are influenced by random outcomes of previous stages Draw three digits at ST2004 Week 722

23. Probability Rules Conditional Prob and Independence Multiplication Rule 23ST2004 Week 7

24. Decomposing with Cond Probs ST2004 Week 724

25. Event Identities: Password ST2004 Week 725 Elementary events and associated probs Pr(Not Dup) via prod rules

26. Applying Cond Probability Rules 26ST2004 Week 7

27. Applying Cond Probability Rules 27ST2004 Week 7 Write down event identities explicitly Justify use of + or  explicitly

28. Bayes Rule & Thinking Backwards ST2004 Week 728 See text, Ch 8.2

29. Bayes Rule & Thinking Backwards ST2004 Week 729

30. Bayes Rule & Thinking Backwards ST2004 Week 730

31. Bayes Rule & Thinking Backwards ST2004 Week 731

32. Probability Distributions and Random Variables Output of a simulation exercise (thought expt) Columns defined random variables Y – Discrete countable list of possible values – Continuousvalues – True/False valuesRandom Var is ‘Event’ Discrete random vars fully described by – 2 listsPoss Values y of Y Associated Probs Pr(Y=y) ST2004 Week 732 Main use of probability

33. Applying Probability Rules – Indep Case 33ST2004 Week 7

34. Applying Probability Rules – Indep Case 34ST2004 Week 7

35. Applying Probability Rules – Indep Case 35ST2004 Week 7 Games are indep

36. Applying Probability Rules – Indep Case 36ST2004 Week 7 Games are indep Scores not indep

37. Conditional Distributions Probabilities must sum to 1! 37ST2004 Week 7

38. Conditional Distributions Probabilities must sum to 1! 38ST2004 Week 7

39. Conditional Distributions and Conditional Expected Values 39ST2004 Week 7

40. Probability and Random Variables Random Variable Y – Univariate or Multivariate – Name – List of Poss Values y Sample space – List of Probs for Events (Y=y)Sum to 1 Events – Composite Events – AND / OR Combinations of Elementary Events Probabilities – Satisfy RulesMult/Add – Conditioned on ‘Knowledge’ 40ST2004 Week 7 (Discrete) Probability distribution