Probability

1. Probability Introduction statistical processes

2. Class 2Readings & Problems • Reading assignment • M & S • Chapter 3 - Sections 3.1 - 3.10 (Probability) • Recommended Problems • M & S Chapter 3 • 1, 20, 25, 29, 33, 57, 75, and83 statistical processes

3. Introduction to Probability • Probability - a useful tool • Inferential statistics • Infer population parameters probabilistically • Stochastic modeling (engineering applications) • Decision analysis • Simulation • Reliability • Statistical process control • Others … statistical processes

4. Development of Probability Theory • Chapter 3 - Introduction to probability • Basic concepts • Chapter 4 - Discrete random variables • What is a random variable??? • What is a discrete random variable??? • Chapter 5 - Continuous random variable • What is a continuous random variable??? • Do not be afraid of random variables!! statistical processes

5. What Is Probability? • Deterministic models • All parameters known with certainty • Stochastic models • One or more parameters are uncertain • May be unknown • Known but may take on more than 1 value • Measure of uncertainty  probability • Probability quantifies uncertainty! statistical processes

6. ProbabilityMost Common Viewpoint • Frequentist view • Probability is relative frequency of occurrence • Most often associated with probability • Adopted in textbook • Probability inherent to physical process • Property of large number () of trials • Examples of applications?? statistical processes

7. ProbabilityAn Alternative Perspective • Bayesian view (aka personalist or subjective) • Many real world applications not amenable to frequentist viewpoint • What is probability of permanent lunar colony by 2015? • What if asked in 1970? • What if asked in 1998? • What if asked in 2004??! • Is probability here a property inherent to physical process? statistical processes

8. Bayesian ProbabilityWhat is key? • What is probability RPI beat Cornell in hockey February 1971? • What is probability RPI beat Cornell in hockey February 1971? • RPI was ECAC champ that year • What is probability RPI beat Cornell in hockey February 1971? • The score was RPI 3, Cornell 1 State of knowledge defines probability statistical processes

9. Flip a coin Heads or tails Frequentist ProbabilityBuilding a Foundation • Experiment • Process of obtaining observations What are examples? • Basic outcome • A simple event • Elemental outcomes What are examples? statistical processes

10. Frequentist ProbabilityDefining Terms • Sample space • Collection of all simple events of experiment • Could be population or sample • Set notation S = { e1, e2, …, en} where, S  sample space ei  possible simple event (outcome) What is sample space for rolling 1 die? What is sample space for rolling 2 dice? statistical processes

11. S0  all men in VA S0 S1 S1  all >6’ men in VA S2 S2  all men >50 in VA Visualizing Sample SpaceVenn Diagram Venn diagram represents all simple events in sample space Is S0 part of a larger sample space? statistical processes

12. Set Terminology • Subsets S0 S1 S1 is a subset of S0 (S0 is a superset of S1) Every point in S1 is in S0 NOTE: S1 could be the same as S0 S0S1 S1 is a strict subset of S0 Every point in S1 is in S0 and S0 S1 statistical processes

13. Defining Probability • p(ei)  probability of ei • Likelihood of ei occurring if perform experiment • Proportion of times you observe ei Recall frequentist viewpoint in word “size” statistical processes

14. Fundamental RulesProbability • If p(ei) = 0  ei will never occur • If p(ei) = 1.0  ei will occur with certainty • Let, E = {ei, …, ej} then, p(E) = p(ei) + … + p(ej) Have 2 dice, find p(toss a 7), p(toss an 11) statistical processes

15. Event Simple events Defining More TermsCompound Events • Let A  event, B  event A  B is the union of A and B (either A or B or both occur) • If C = A  B then A  C, and B  C • If A event you toss 7, B event you toss 11, and C = A  BWhat is C Recall E = {ei, …, ej} statistical processes

16. B A A A B B Visualizing Union of SetsVenn Diagrams C = A  B statistical processes

17. S0  all men in VA S0 S1 S1  all >6’ men in VA S2 S2  all men >50 in VA Let C = S1 S2 What does C represent?? Defining More TermsIntersection of Sets statistical processes

18. A B A and B are mutually exclusive  A  B =  (the null set) Intersection of SetsDice Example • Consider toss of 2 dice, let A = event you toss a 7 B = event you toss an 11 C = A  B • Draw Venn Diagram showing C statistical processes

19. S ~A A ComplementarityA Useful Concept • Let A be an event • then ~A is event that A does not occur ~A is the complement of A ~A read as “not A” also shown as Ac, A Ac and A read as “the complement of A” • p(A) + p(~A) = 1.0 statistical processes

20. Conditional ProbabilityStrings Attached • Are these likely the same? • p(person in VA > 6’ tall) • p(person in VA > 6’ tall given person is a man) • Former is an unconditional probability • Latter is a conditional probability • Probability of one event given another event has occurred • Formal nomenclature p(A  B) statistical processes

21. S A  B A B Conditional Probability Formula statistical processes

22. Conditional ProbabilitiesExample Problem • Study of SPC success at plants A = plant reports success; B = plant reports failure C = plant has formal SPC; D = plant has no formal SPC What are: p(AC)? p(C)? p(AC)? p(BC)? statistical processes

23. A  B S A B Additive Rule of ProbabilityIntuitive Result Additive Rule for Mutually Exclusive Events 1) p(AB)=0 2) p(AB) = p(A) + p(B) What if A & B are mutually exclusive? statistical processes

24. Exercise • Deck of 52 playing cards • What is p(picking a heart or a jack)??? statistical processes

25. Exercise • Same deck of 52 cards • What is p(jack  card is a heart)? • What is p(heart  card is a jack)? • Your results should make sense statistical processes

26. Multiplicative Rule • Recall, conditional probability formula p(A  B) = p(A  B) / p(B) • Multiplicative Rule p(A  B) = p(B) p(A  B) = p(A) p(B  A) • Remember: • Additive rule applies to p(A  B) • Multiplicative rule applies to p(A  B) statistical processes

27. Special Case of Conditional Probability:What if the Conditions Do Not Matter? • What is p(toss head  previous toss was tail)? • p(toss head  previous toss was tail) = p(toss head) • Independent events defined as p(A  B) = p(A) p(B  A) = p(B) • Multiplicative rule for independent events p(A  B) = p(B) p(A) = p(A) p(B) statistical processes

28. Confirming IndependenceDo Not Trust Intuition • Can Venn Diagrams illustrate independence? • No! • Unlike mutually exclusive events • How to demonstrate A & B are independent? • See if p(A  B) = p(B) p(A) • See Examples 3.16 & 3.17, assigned problem 3.24 • Not through Venn Diagram • Are mutually exclusive events independent? • No! p(A  B) = 0  p(B) p(A) statistical processes

29. Counting Rules • Counting rules • Finding number of simple events in experiment • aka Combinatorial Analysis • Why would this be important? • Most important rules • Permutations • Combinations statistical processes

30. PermutationsRepresentative Application • You are employer • 2 open positions, J1 and J2 • 5 applicants {A, B, C, D, E} for either job • How many ways to fill positions?? statistical processes

31. B C A J2 D E B J1 C D E PermutationsVisualizing Problem And so forth. Total of 20 possibilities. Decision tree representation Tool for sequential combinatorial analysis Decisions to fill open jobs statistical processes

32. Permutation Formula • Is A getting J1 same as A getting J2? • Order important • Basic distinction of permutation problems • Permutation formula • N! said as “N factorial” • N! = (N)(N-1) … (1) • 0! = 1 • Multiplicative Rule: Basis of permutation formula statistical processes

33. Permutation RuleMore Formal Definition • Given SN { ej  j = 1, …, N} • Select subset of n members from SN • Order is important statistical processes

34. A total of 10 combinations! CombinationsOrder Is Not Important • Suppose J1 and J2 were the same • Order not important • How would you enumerate combinations? • Choose A for J1 • AB, AC, AD, AE • Choose B for J1 • BC, BD, BE • Choose C for J1 • CD, CE • Choose D for J1 • DE statistical processes

35. Combinations Rule More Formal Definition • Given SN { ej  j = 1, …, N} • Select subset of n members from SN • Order is not important • Effectively a sample from SN statistical processes

36. Subset with n members SN Set with N members Subset with (N-n) members Combinations RuleDifferent Perspective How many ways can you break up set SN into two subsets: one with n and the other with (N-n) members? statistical processes

37. Original set One of the subsets The second subset Interpreting theCombinations Rule Can you generalize breaking up into > 2 subsets??? statistical processes

38. Note special case when k=2 Partitions RuleBreaking Set into k Subsets • Given SN { ej  j = 1, …, N} • Select k subsets from SN • Each subset has n1, n2, … , nk members • Order is not important statistical processes

39. Partitions RuleA Personal Experience • Have 55 kids, how many different teams of 11 players each? statistical processes

40. Useful Excel FunctionsWhen You Work With Real Data MEAN MEDIAN MODE PERMUT PERCENTILE FACT STDEV STDEVP VAR VARP DEVSQ Statistical Special Functions Excel statistical processes