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Applying Probability

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

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Applying Probability

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  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 ST2004 Week 7

  2. Probability 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. • Prob Rules Week 7 • Basic in text, Ch2.2 • Conditional ProbBayes 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 Dist Week 9 • Ch 5 Normal dist • Ch 10 ST2004 Week 7

  3. Problems • Dice: Seek prob dist of M2,S2 ,M3,S3 ,Mk,Sk • Later E(S2) Var(S2) etc • Mini-league: Seek prob dist of (NA, NB, NC) when • Pr( A beats B)=2 Pr(B beats A)  Pr( A beats B)=? • Pr( A beats B)=pAB; similarly pBC, pAC • Later E(NA),Var(NA) and E(NA|NC=0),Var(NA|NC=0) ST2004 Week 7

  4. 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 y1, y2..yk(sample space; exhaustive list) Events such as (Y= y)

  5. 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♥) Event Algebra

  6. 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 NA , 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” Event Identities

  7. Event Identities ST2004 Week 7

  8. Event Identities ST2004 Week 7

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

  10. Coins/Dice/Cards ST2004 Week 7

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

  12. Event Identities: Password Elementary events and associated probs Pr(Dup) via addition rules ST2004 Week 7

  13. Conditional Probability ST2004 Week 7

  14. Probability RulesConditional Prob and Independence Multiplication Rule ST2004 Week 7

  15. Decomposing with CondProbs ST2004 Week 7

  16. Applying Cond Probability Rules ST2004 Week 7

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

  18. Bayes Rule & Thinking Backwards See text, Ch 8.2 ST2004 Week 7

  19. Bayes Rule & Thinking Backwards ST2004 Week 7

  20. Bayes Rule & Thinking Backwards ST2004 Week 7

  21. Probability Distributionsand Random Variables Main use of probability • Output of a simulation exercise (thought expt) • Columns defined random variables Y • Discrete countable list of possible values • Continuous values • True/False values Random Var is ‘Event’ • Discrete random vars fully described by • 2 lists Poss Values y of Y Associated Probs Pr(Y=y) ST2004 Week 7

  22. Applying Probability Rules – Indep Case ST2004 Week 7

  23. Applying Probability Rules – Indep Case ST2004 Week 7

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

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