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
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.
ST 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
Cards and Queens Simulation Probability 4ST2004 Week 7
Mini-League Probs 5ST2004 Week 7
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
ST 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
ST 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
ST 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
Event Identities 10ST2004 Week 7
Event Identities: Coin Toss ST2004 Week 711
Event Identities: Password ST2004 Week 712
Event Identities 13ST2004 Week 7
Event Identity etc 14ST2004 Week 7
Event Identity etc 15ST2004 Week 7
Event Identities for Random Vars 16ST2004 Week 7
Probability Rules Addition Rule 17ST2004 Week 7 Plus real world knowledge
Coins/Dice/Cards 18ST2004 Week 7
Applying Prob Rules Generalisation of Addition Rule 19ST2004 Week 7
Event Identities: Password ST2004 Week 720 Elementary events and associated probs Pr(Dup) via addition rules
Conditional Probability 21ST2004 Week 7
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
Probability Rules Conditional Prob and Independence Multiplication Rule 23ST2004 Week 7
Decomposing with Cond Probs ST2004 Week 724
Event Identities: Password ST2004 Week 725 Elementary events and associated probs Pr(Not Dup) via prod rules
Applying Cond Probability Rules 26ST2004 Week 7
Applying Cond Probability Rules 27ST2004 Week 7 Write down event identities explicitly Justify use of + or explicitly
Bayes Rule & Thinking Backwards ST2004 Week 728 See text, Ch 8.2
Bayes Rule & Thinking Backwards ST2004 Week 729
Bayes Rule & Thinking Backwards ST2004 Week 730
Bayes Rule & Thinking Backwards ST2004 Week 731
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
Applying Probability Rules – Indep Case 33ST2004 Week 7
Applying Probability Rules – Indep Case 34ST2004 Week 7
Applying Probability Rules – Indep Case 35ST2004 Week 7 Games are indep
Applying Probability Rules – Indep Case 36ST2004 Week 7 Games are indep Scores not indep
Conditional Distributions Probabilities must sum to 1! 37ST2004 Week 7
Conditional Distributions Probabilities must sum to 1! 38ST2004 Week 7
Conditional Distributions and Conditional Expected Values 39ST2004 Week 7
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