Chapter 4

Basic ProbabilityStatistics for ManagersUsing Microsoft Excel 4th Edition

Chapter GoalsAfter completing this chapter, you should be able to: Explain basic probability concepts and definitionsUse contingency tables to view a sample spaceApply common rules of probabilityCompute conditional probabilitiesDetermine whether events are statistically independentUse Bayes Theorem for conditional probabilities

Important TermsProbability the chance that an uncertain event will occur (always between 0 and 1)Event Each possible type of occurrence or outcomeSimple Event an event that can be described by a single characteristicSample Space the collection of all possible events

Assessing ProbabilityThere are three approaches to assessing the probability of an uncertain event:1. a priori classical probability

2. empirical classical probability

3. subjective probability an individual judgment or opinion about the probability of occurrence

Sample SpaceThe Sample Space is the collection of all possible eventse.g. All 6 faces of a die:

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

EventsSimple eventAn outcome from a sample space with one characteristice.g.: A red card from a deck of cardsComplement of an event A (denoted A)All outcomes that are not part of event Ae.g.: All cards that are not redJoint eventInvolves two or more characteristics simultaneouslye.g.: An ace that is also red from a deck of cards

Visualizing EventsContingency Tables

Tree Diagrams Red 2 24 26 Black 2 24 26Total 4 48 52 Ace Not Ace TotalFull Deck of 52 CardsRed CardBlack CardNot an AceAceAceNot an Ace Sample SpaceSample Space224224

Special EventsMutually exclusive eventsEvents that cannot occur together

example:

A = queen of diamonds; B = queen of clubs

Events A and B are mutually exclusive

Special EventsCollectively exhaustive eventsOne of the events must occur The set of events covers the entire sample space

example: A = aces; B = black cards; C = diamonds; D = hearts

Events A, B, C and D are collectively exhaustive (but not mutually exclusive an ace may also be a heart)Events B, C and D are collectively exhaustive and also mutually exclusive(continued)

ProbabilityProbability is the numerical measure of the likelihood that an event will occurThe probability of any event must be between 0 and 1

The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1CertainImpossible.5100 P(A) 1 For any event A

Computing Joint and Marginal ProbabilitiesThe probability of a joint event, A and B:

Computing a marginal probability:

Where B1, B2, , Bk are k mutually exclusive and collectively exhaustive events

Joint Probability ExampleP(Red and Ace)BlackColorTypeRedTotalAce224Non-Ace242448Total262652

Marginal Probability ExampleP(Ace)BlackColorTypeRedTotalAce224Non-Ace242448Total262652

Joint Probabilities Using Contingency Table P(A1 and B2)P(A1)TotalEventP(A2 and B1)P(A1 and B1)EventTotal1Joint ProbabilitiesMarginal (Simple) Probabilities A1 A2B1B2 P(B1) P(B2)P(A2 and B2)P(A2)

General Addition RuleP(A or B) = P(A) + P(B) - P(A and B)General Addition Rule:If A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified:P(A or B) = P(A) + P(B)

General Addition Rule ExampleP(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52Dont count the two red aces twice!BlackColorTypeRedTotalAce224Non-Ace242448Total262652

Computing Conditional ProbabilitiesA conditional probability is the probability of one event, given that another event has occurred:Where P(A and B) = joint probability of A and B P(A) = marginal probability of AP(B) = marginal probability of B

Conditional Probability ExampleWhat is the probability that a car has a CD player, given that it has AC ?

i.e., we want to find P(CD | AC)Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

Conditional Probability ExampleNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.(continued)

Conditional Probability ExampleNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%.(continued)

Using Decision TreesHas ACDoes not have ACHas CDDoes not have CDHas CDDoes not have CDP(AC)= .7P(AC)= .3P(AC and CD) = .2P(AC and CD) = .5P(AC and CD) = .1P(AC and CD) = .2AllCarsGiven AC or no AC:

Using Decision TreesHas CDDoes not have CDHas ACDoes not have ACHas ACDoes not have ACP(CD)= .4P(CD)= .6P(CD and AC) = .2P(CD and AC) = .2P(CD and AC) = .1P(CD and AC) = .5AllCarsGiven CD or no CD:(continued)

Statistical IndependenceTwo events are independent if and only if:

Events A and B are independent when the probability of one event is not affected by the other event

Multiplication RulesMultiplication rule for two events A and B:Note: If A and B are independent, thenand the multiplication rule simplifies to

Bayes Theoremwhere:Bi = ith event of k mutually exclusive and collectively exhaustive eventsA = new event that might impact P(Bi)

Bayes Theorem ExampleA drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?

Let S = successful well U = unsuccessful wellP(S) = .4 , P(U) = .6 (prior probabilities)Define the detailed test event as DConditional probabilities:P(D|S) = .6 P(D|U) = .2Goal is to find P(S|D)Bayes Theorem Example(continued)

So the revised probability of success, given that this well has been scheduled for a detailed test, is .667Bayes Theorem Example(continued)Apply Bayes Theorem:

Given the detailed test, the revised probability of a successful well has risen to .667 from the original estimate of .4Bayes Theorem ExampleSum = .36 = P(D)(continued)

EventPriorProb.Conditional Prob.JointProb.RevisedProb.S (successful).4.6.4*.6 = .24.24/.36 = .667U (unsuccessful).6.2.6*.2 = .12.12/.36 = .333

Chapter SummaryDiscussed basic probability conceptsSample spaces and events, contingency tables, simple, joint, marginal and conditional probabilities Examined basic probability rulesGeneral addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive eventsDefined conditional probabilityStatistical independence, marginal probability, decision trees, and the multiplication ruleDiscussed Bayes theorem