Probability Probability – Long Run Relative Frequency of an Event Under a Constant Cause System. Classical Probability Formula P(A) = # Outcomes which Favor A Total # Outcomes (All Outcomes are equally likely) Personal or Subjective Probability Probability that I will get an A Probability that I will get Married this year Probability that I will Die this year

Random Experiment – An Experiment whose Outcomes are Determined by Probability Toss a Coin Select an Employee Roll a Die Inspect an Item Draw a Card Elementary Events – Outcomes which cannot be Decomposed into other Outcomes Sample Space – A Collection of all possible Outcomes for an Experiment (Mutually Exclusive, Exhaustive) Event – Any Event is a Collection of Elementary Events A(Even Roll) = {2,4,6} B(High Roll) = {5,6}

A B A B Compound Events – Conjunction of two Events Intersection – The Intersection of Events A and B is composed of all Elementary Events Common to Both A and B and denoted by: A and B A ∩ B A and B = {6} Union – The Union of Events A and B is composed of those Elementary Events Common to A Only, those Common to B Only, and those Common to Both A and B and denoted by: A or B A U B A or B = {2,4,5,6} B A B A

Mutually Exclusive – 2 Events are Mutually Exclusive if the occurrence of one Event precludes the occurrence of the other Event Independence – Two Events are Independent if the occurrence of one Event has no effect on the occurrence of the other Event, otherwise they are Dependent. Complement – The Complement of Event A, denoted by A, is composed of all outcomes not in A B A

Probability Postulates: SS = {e1, e2, e3, e4, e5, e6} 0 ≤ P( ej ) ≤ 1 P(e1) + P(e2) + P(e3) + P(e4) + P(e5) + P(e6) = 1 If A = {ei, ej, ek), then P(A) = P(ei) + P(ej) + P(ek) Ex: Choose two people from 3 – Women and 2 – Men Women = {1,2,3} Men = {4,5} e1 = {1,2} e5 = {2,3} e8 = {3,4} e10 = {4,5} e2 = {1,3} e6 = {2,4} e9 = {3,5} e3 = {1,4} e7 = {2,5} e4 = {1,5} (2W) = {e1,e2,e5} (2M) = {e10} (WandM) ={e3,4,e6,e7,e8,e9}

Mathematical Counting Rules: M•N Rule – If there are M first outcomes and N second outcomes, then there are M*N Total outcomes Roll a Pair of Dice - 6 * 6 = 36 outcomes License Plate – ABC 123 26*26*26*10*10*10 = 17,576,000 12 A 3456 26*10*10*10*10*10*10 = 26,000,000 Combinations - # of ways of selecting r items from n total items without regard to order of selection n! = n(n-1)(n-2)(n-3)…(3)(2)1 Choose 2 people from 5 people -

Draw 5 Cards from a 52 Card Deck - 52C5 =

Ex: Select 4 Items from 20 Items of which 5 are Defective

2100 4200 700 900 1800 300 Bivariate Frequency Distribution Gender\Age <35yr 35-50yr >50yr 2100 4200 700 900 1800 300 Male Female Probability Distribution Gender\Age <35yr 35-50yr >50yr Male Female Marginal Probability – Joint Probability -

Addition Rule – P( A or B ) = P(A) + P(B) – P( A and B ) Conditional Probability – Probability of Event A Given that Event B is Certain

Independence – Events A and B are Independent If P(A|B) = P(A) and P(B|A) = P(B) ; otherwise Dependent _ Complement Rule – P( A ) + P( A ) = 1 P( A ) = 1 - P( A ) _ Game - Toss a Coin until you get a Tail # Tosses 1 2 3 4 5 6 7 Outcome T HT HHT HHHT HHHHT HHHHHT HHHHHHT Probability 1/2 1/4 1/8 1/16 1/32 1/64 1/128 P(X > 1 Toss) = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + etc P( X > 1 Toss) = 1 – P( 1 Toss )

Multiplication Rule - P( A and B ) = P( A ) * P( B | A ) Probability Formulas: 1) 2) 3) 4) _ P( A ) = 1 - P( A ) P( A or B ) = P(A) + P(B) – P( A and B ) P( A and B ) = P( A ) * P( B | A )

Mosaic Plot of Categorical Data

Survival Plot for Donner Party

Example – 4.35 10 20 15 5 30

Example – 4.41

Bayes Theorem - Revision of Probability with Additional Information P(A) – Prior Prob of Event (Subjective) B – Additional Info (Objective) P(A|B) – Revised Prob of the Event Bayes Rule Example: A – Event that You are an A or B Student , P(A) = .60 B – Score >= 80% on 1st Test P(B|A) = .95 P(B|A) = .15 P(A|B) = _

Other Solution Methods: Prior Likelihood Joint Revised State P(Ai) P(B|Ai) P(Ai)*P(B|Ai) P(Ai|B) A or B C or D

Example – 4.34 Birthday Match ? What is the Probability that at least two people in the Room have the Same Birthday?

Birthday Match ? What is the Probability that at least two people in the Room have the Same Birthday? P(Match) = 1 - P(No Match) P(No Match) = (365/365)(364/365)(363/365)(362/365)(361/365)…etc # People P(No Match) P(Match) 10 .883 .117 20 .589 .411 23 .493 .507 30 .294 .706 40 .109 .891 50 .030 .970