Basic Probability And Probability Distributions

Probability Theory Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Topics Basic Probability Concepts:Sample Spaces and Events, Simple Probability, and Joint Probability, Conditional Probability Bayes Theorem Probability Distribution

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Sample Spaces Collection of all Possible Outcomes e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Events Simple Event: Outcome from a Sample Space with 1 Characteristice.g. A Red Card from a deck of cards. Joint Event: Involves 2 Outcomes Simultaneouslye.g. An Ace which is also a Red Card from a deck of cards. Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Visualizing Events Contingency Tables

Ace Not Ace Total Red 2 24 26 Black 2 24 26Total 4 48 52 Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Simple EventsThe Event of a Happy Face

There are 5 happy faces in this collection of 18 objects Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#

Joint EventsThe Event of a Happy Face AND Light Colored3 Happy Faces which are light in color Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Special Events Null event Club & diamond on 1 card draw Complement of event For event A, All events not In A: Null Event

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#

3 Items: 3 Happy Faces Given they are Light ColoredDependent or Independent EventsThe Event of a Happy Face GIVEN it is Light ColoredE = Happy FaceLight Color Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Contingency TableA Deck of 52 CardsAceNot anAceTotalRedBlackTotal224224262644852Sample SpaceRed Ace Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#2500Contingency Table2500 Employees of Company ABCAgreeNeutralOpposed | Total MALEFEMALETotal 900200300 100400 | 1500 600 | 1000 12003001000 | Sample Space Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#

Tree DiagramEvent PossibilitiesRed CardsBlack CardsAceNot an AceAceNot an Ace Full Deck of Cards Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#ProbabilityProbability is the numerical measure of the likelihood that the event will occur.Value is between 0 and 1.Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1.CertainImpossible.510 Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Computing Probability The Probability of an Event, E:

Each of the Outcome in the Sample Space equally likely to occur.

e.g. P( ) = 2/36(There are 2 ways to get one 6 and the other 4)P(E) =Number of Event OutcomesTotal Number of Possible Outcomes in the Sample Space=XT Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Computing Joint ProbabilityThe Probability of a Joint Event, A and B: e.g. P(Red Card and Ace) P(A and B) Number of Event Outcomes from both A and BTotal Number of Possible Outcomes in Sample Space==

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#P(A2 and B1)P(A1 and B1)EventEventTotal Total1Joint Probability Using Contingency TableJoint ProbabilityMarginal (Simple) ProbabilityP(A1)A1A2B1B2 P(B1) P(B2)P(A1 and B2)P(A2 and B2)P(A2) Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-# Computing Compound ProbabilityThe Probability of a Compound Event, A or B: e.g. P(Red Card or Ace)

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#2500Contingency Table2500 Employees of Company ABCAgreeNeutralOpposed | Total MALEFEMALETotal 900200300 100400 | 1500 600 | 1000 12003001000 | Sample Space Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposalThe pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal .. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal .. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56

5. Are Gender and Opinion (statistically) independent?

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussionCalculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal .. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56

5. Are Gender and Opinion (statistically) independent? For Opinion and Gender to be independent, the joint probability of each pair of A events (GENDER) and B events (OPINION) should equal the product of the respective unconditional probabilities.clearly this does not hold..check, e.g., the prob. Of MALE and IN FAVOR against the prob. of MALE times the prob. of IN FAVOR they are not equal.900/2500 does not equal 1500/2500 * 1200/2500

P(A1 and B1)P(B2)P(B1)P(A2 and B2)P(A2 and B1)EventEventTotal Total1Compound ProbabilityAddition RuleP(A1 and B2)P(A1)A1A2B1B2P(A2)P(A1 or B1 ) = P(A1) +P(B1) - P(A1 and B1)For Mutually Exclusive Events: P(A or B) = P(A) + P(B) Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-# Computing Conditional ProbabilityThe Probability of Event A given that Event B has occurred: P(A B) = e.g. P(Red Card given that it is an Ace) =

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#BlackColorTypeRedTotalAce224Non-Ace242448Total262652Conditional Probability Using Contingency TableConditional Event: Draw 1 Card. Note Kind & Color

Revised Sample Space Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-# Conditional Probability and Statistical Independence

Conditional Probability:P(AB) =P(A and B) = P(A B) P(B)Multiplication Rule:= P(B A) P(A) Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-# Conditional Probability and Statistical Independence (continued)Events are Independent:P(A B) = P(A)Or, P(A and B) = P(A) P(B) Events A and B are Independent when the probability of one event, A is not affected by another event, B.Or, P(B A) = P(B) Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Bayes Theorem

P(Bi A) =

Adding up the parts of A in all the BsSame Event Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Bayes TheoremGiven a hypothesis H and an event E

P(H|E) = P(E|H) x P(H) P(E)

Where P(E) = P(E|H) x P(H) + P(E|H) x P(H)

14The probability for the truth of a hypothesis in the light of new evidence E is given by the probability of H being correct given piece of evidence E times the prior probability of the hypothesis being correct all over the probability that the evidence will occur. The probability the evidence will occur is given by P(E|H)xP(H) plus the probability of the alternative hypothesis being true given the evidence times the prior probability of the alternative hypothesis. Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#ExampleYou have a physical complaint which is one of the symptoms of a rare and unpleasant disease. 1 in 500 people have the disease. The probability of having the physical symptom if you have the disease is 0.7. The probability of having the symptom without also having the disease is 0.01.How likely is it that you are suffering from the disease?15Note that humans tend to underestimate the effect of base-rate information when making these kind of judgements (I.e. underestimate the effect of the prob. of the prior prob. of having the the disease).

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#ExampleP(E) = P(E|H) x P(H) + P(E|H) x P(H)P(E) = 0.7 x 0.002 + 0.01 x 0.998 = 0.01138

P(H|E) = P(E|H) x P(H) P(E) P(H|E) = 0.7 x 0.002 = 0.123 0.01138

16Prob of symptom E given disease H is 0.7Prob of disease H is 1/500 = .002Prob of the symptom E given not disease H is 0.01Prob of not having the disease is 1-0.002 = .998

Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#Probability Distribution ModelsProbability DistributionsContinuousDiscrete Statistics for Managers Using Microsoft Excel, 2/e 1999 Prentice-Hall, Inc.Chapter 4Instructor Notes4-#