Probability 1 Probability Long Run Relative Frequency of

Probability 1) Probability – Long Run Relative Frequency of an Event Under a Constant Cause System. 2) Classical Probability Formula P(A) = # Outcomes which Favor A Total # Outcomes (All Outcomes are equally likely) 3) Personal or Subjective 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}

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 B 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 B A or B A U B A or B = {2, 4, 5, 6}

Mutually Exclusive – 2 Events are Mutually Exclusive if the occurrence of one Event precludes the occurrence of the other Event A B 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

Probability Postulates: SS = {e 1, e 2, e 3, e 4, e 5, e 6} 1) 0 ≤ P( ej ) ≤ 1 2) P(e 1) + P(e 2) + P(e 3) + P(e 4) + P(e 5) + P(e 6) = 1 3) 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} e 1 = {1, 2} e 5 = {2, 3} e 8 = {3, 4} e 2 = {1, 3} e 6 = {2, 4} e 9 = {3, 5} e 3 = {1, 4} e 7 = {2, 5} e 10 = {4, 5} e 4 = {1, 5} (2 W) = {e 1, e 2, e 5} (2 M) = {e 10} (Wand. M) ={e 3, 4, e 6, e 7, e 8, e 9}

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 = 26, 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 - 52 C 5 Draw 13 Cards from a 52 Card Deck - = 52 C 13 =

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

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

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 Outcome T HT HHHT HHHHHT HHHHHHT 1/4 1/8 1/16 1/32 Probability 1/2 6 1/64 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 ) 7 1/128

Multiplication Rule - P( A and B ) = P( A ) * P( B | A ) Probability Formulas: _ 1) P( A ) = 1 - P( A ) 2) P( A or B ) = P(A) + P(B) – P( A and B ) 3) 4) 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 15

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 1 st Test P(B|A) =. 95 P(B|A) =. 15 P(A|B) =

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

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) = (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