TheThe BinomialBinomial DistributionDistribution

SummerSummer 20032003 InternetInternet BubbleBubble “Several“Several industryindustry expertsexperts believebelieve thatthat 30%30% ofof internetinternet companiescompanies willwill runrun outout ofof cashcash inin 66 monthsmonths andand thatthat thesethese companiescompanies willwill findfind itit veryvery hardhard toto securesecure additionaladditional funding”funding” AssociationAssociation forfor ComputingComputing MachineryMachinery,, August,August, 20012001

PickPick 1010 companiescompanies atat random.random. HowHow manymany wouldwould wewe expectexpect toto runrun outout ofof cashcash inin 66 months?months? WhatWhat isis p(4p(4 runrun outout ofof cashcash inin 66 months)?months)? WhatWhat isis p(2p(2 runrun out…)?out…)? WhatWhat isis p(allp(all 1010 runrun out…)?out…)?

15.063 Summer 2003 2 BinomialBinomial DistributionDistribution

Derived from theory, not from experience • An experiment consists of n “trials” • Each trial results in : yes or no (“binomial” means “2 names” or “2 labels”) • Trials are independent of each other • Each trial has same probability: success p, failure 1-p

r.v. X = # successes in n trials

Random variable X has binomial distribution with parameters n and p “Binomial(n,p)”

15.063 Summer 2003 3 AnAn ExampleExample “Experiment”“Experiment” Select, randomly, one team of 3 students from a large pool of students composed of 70% US, and 30% International. r.v. X = number of international students on the three student team selected randomly. •n= 3 trials • define: “yes”: picking an international student “no”: picking a US student • P(yes) = p = 0.30; P(no) = 1 - p = 0.70

X is a Binomial (n=3 , p=0.30) (compare to experiment 2 in which we flipped three coins with p(H) = 0.3 and looked at the total # of heads) 15.063 Summer 2003 4 ProbabilityProbability treetree andand probabilityprobability distributiondistribution forfor r.v.r.v. XX (total(total ## IntInt students)students)

Ou t c o m e X (# I n t) Pr o b a b i l In t I n t I n t I n t 3 0. 027 In t 0 .3 0. 3 US I n t I n t U S 2 0 . 063 In t 0 .7 0. 3 In t I n t U S I n t 2 0 . 063 US 0 . 3 X p( X ) 0. 7 0 0 . 343 U S 1 0 . 441 I n t U S U S 1 0. 147 2 0 . 189 0. 7 3 0 . 027 1. 000 In t U S I n t I n t 2 0 . 063 In t 0 .3 0. 3 US U S I n t U S 1 0. 147 US 0 . 7 0. 7 In t U S U S I n t 1 0. 147 US 0 . 3 0. 7 US US US US 0 0 . 3 4 3 0. 7 1 . 000

15.063 Summer 2003 5 BinomialBinomial ProbabilityProbability DistributionDistribution FormulaFormula

Let X be Binomial(n, p): The probability of having x “yes” events (also called “successes” in n trials is

P(X=x) = n! px (1-p) n-x x!(n − x)! (where x! = x(x-1)(x-2)…1, and 0! = 1)

Check in Example 1: n=3 , p=0.3

P(X=1) = 3! (0.3) (0.7) 2 = 0.441 1!/(3-1)!

15.063 Summer 2003 6 InternetInternet BubbleBubble AgainAgain

WhatWhat isis thethe p(4p(4 ofof 1010 runrun outout ofof cash)?cash)? P(X=x)P(X=x) == n!n! ppx(1(1--p)p)n-x x!(nx!(n--x)!x)! P(X=4)P(X=4) == 10!10! .3.34 (.7(.76)) == .20.20 4!6!

P(X=10)P(X=10) == ??

15.063 Summer 2003 7 ExcelExcel CalculatesCalculates BinomialBinomial ProbabilitiesProbabilities

UseUse thethe functionfunction wizardwizard iconicon oror thethe Insert_FunctionInsert_Function CommandCommand toto choosechoose thethe StatisticalStatistical functionfunction BINOMDISTBINOMDIST.. ThisThis functionfunction calculatescalculates eithereither thethe individualindividual binomialbinomial probabilityprobability P(X=Value)P(X=Value) oror thethe cumulativecumulative binomialbinomial probabilityprobability P(X<=Value).P(X<=Value).

15.063 Summer 2003 8 Mean,Mean, VarianceVariance andand SDSD forfor BinomialBinomial ProbabilityProbability DistributionDistribution (r.v.)(r.v.)

Let X be Binomial(n, p): The probability of having x successes in n trials is:

n! P(X=x)= x n-x x!(n − x)! p (1-p) (where x! = x(x-1)(x-2)…1, and 0! = 1)

E(X) = np = 3* 0.3 = 0.9 Calculations shown for Binomial (n=3, p=0.3) VAR(X) = np(1-p) = 3* 0.3 * 0.7 = 0.63 Note: this is equivalent to SD(X) = np(1-p) = 0.794 counting success = 1 and failure = 0

15.063 Summer 2003 9 Another example:

A laser production facility is known to have a 75% yield; that is, 75% of the lasers manufactured by the facility pass the quality test. Suppose that today the facility is scheduled to produce 15 lasers.

a) What is the expected number of lasers to pass the test (the mean)? b) What is the variance? c) What is the standard deviation? d) What is the probability that at least 14 lasers will pass the quality test?

15.063 Summer 2003 10 LaserLaser ProductionProduction

WhatWhat isis thethe mean?mean? E(X)E(X) == npnp == 15*.7515*.75 == 11.2511.25 laserslasers WhatWhat isis thethe variance?variance? Var(XVar(X)) == np(1np(1--p)p) == 15*.75*.2515*.75*.25 == 2.8122.812 WhatWhat isis thethe standardstandard deviation?deviation? SD(X)SD(X) == sqrtsqrt (Variance)(Variance) == 1.6771.677 laserslasers WhatWhat isis thethe pp ofof 1414 oror moremore passing?passing? P(14P(14 oror more)more) == p(14)p(14) ++ p(15)p(15) == .067.067 ++ .013.013 == .08.08

15.063 Summer 2003 11 AcquiringAcquiring AA CompanyCompany DecisionDecision You represent Company A (the Acquirer), which is considering acquiring Company T (the Target) by means of a cash offer for 100% of Company T shares. You are unsure what price to offer because the value of Company T depends on the outcome of a major oil exploration project. If the project fails, Company T will be worth nothing, but if it succeeds, shares of Company T could be worth up to $100. All share values between $0 and $100 are considered equally likely. By all estimates, whatever the value of Company T under current management, it will be worth 50% more under the management of Company A. The Board of Directors wants your recommendation today for how much to offer per share for Company T. You do not know the outcome of the exploration project, however, you expect Company T to delay their decision about accepting your offer until after they know the results of the project. Company T will accept any offer above their true value and reject any offer below. There will be no negotiation or opportunity to make a second offer. What do you recommend as a take-it-or- leave-it price for Company T? $ _____ per share

15.063 Summer 2003 12 SettingSetting UpUp thethe ProblemProblem

WhatWhat isis ourour goalgoal?? ToTo maximizemaximize expectedexpected profit.profit. WhatWhat areare thethe optionsoptions?? MakeMake anan offeroffer ofof somewheresomewhere betweenbetween $0$0 andand $100/share.$100/share. IfIf youyou havehave troubletrouble settingsetting upup thethe continuouscontinuous problem,problem, trytry makingmaking itit discrete,discrete, e.g.,e.g., whatwhat ifif II offeredoffered $20,$20, $40,$40, $60,$60, $80?$80? WhatWhat areare thethe outcomesoutcomes?? ProfitProfit == RevenueRevenue –– Costs.Costs. RevenueRevenue == 1.51.5 ** BookBook Value.Value. CostsCosts == Offer.Offer. SoSo ProfitProfit == 1.51.5 BVBV –– O.O. WhatWhat isis E(BV)?E(BV)? AndAnd whatwhat areare thethe probabilitiesprobabilities??

15.063 Summer 2003 13 ProbabilitiesProbabilities

WeWe areare toldtold therethere isis aa uniformuniform distributiondistribution betweenbetween $0$0 andand $100.$100. Therefore,Therefore, isis thethe expectedexpected valuevalue == (0(0 ++ 100)/2100)/2 == $50/share?$50/share? No.No. TheThe sellerseller willwill notnot acceptaccept anyany offeroffer lessless thanthan thethe bookbook value.value. Therefore,Therefore, ifif mymy offeroffer isis accepted,accepted, thethe bookbook valuevalue mustmust bebe uniformuniform betweenbetween $0$0 andand Offer.Offer. E(BV)E(BV) isis thereforetherefore (0(0 ++ Offer)/2Offer)/2 == .5.5 OfferOffer !!!! And,And, ProfitProfit == 1.51.5 BVBV –– OfferOffer == 1.5*.51.5*.5 OfferOffer –– OfferOffer == --.25.25 OfferOffer

15.063 Summer 2003 14 UniformUniform DistributionsDistributions

$0 Offer = $60 $100

15.063 Summer 2003 15 ConditionalConditional ProbabilitiesProbabilities TheThe keykey toto thethe AcquiringAcquiring AA CompanyCompany decisiondecision isis toto realizerealize thatthat thethe sellerseller isis makingmaking aa decisiondecision conditionalconditional onon receivingreceiving anan acceptableacceptable offeroffer IfIf youyou offeroffer $60/share,$60/share, thethe offeroffer willwill bebe acceptedaccepted onlyonly ifif thethe valuevalue isis <=<= $60/share$60/share YourYour expectedexpected bookbook valuevalue isis thereforetherefore conditionalconditional onon youryour offer!offer! IfIf youyou offeroffer $60/share,$60/share, thethe expectedexpected bookbook valuevalue isis notnot $50/share,$50/share, butbut ratherrather $30/share$30/share (all(all valuesvalues betweenbetween $0$0 andand $60$60 equallyequally likely)likely) YourYour E(profit)E(profit) isis alwaysalways <0,<0, soso don’tdon’t bidbid !!

15.063 Summer 2003 16 YourYour AnswersAnswers

37%37% answeredanswered $50/share,$50/share, whichwhich isis thethe EMVEMV ofof T;T; ignoresignores thethe conditionalconditional situationsituation 19%19% bidbid betweenbetween $51$51--$75$75 whichwhich triestries toto givegive eacheach sideside somesome profitprofit 9%9% gavegave thethe correctcorrect answeranswer ($0,($0, oror don’tdon’t makemake aa bid)bid) EvenEven $1/share$1/share bidbid willwill loselose anan expectedexpected 2525 centscents perper shareshare (times(times millionsmillions ofof shares)shares)

15.063 Summer 2003 17 Next Lecture Question: Given the choice to invest $1 million in these three firms, which one(s) would you choose?

Expected Return Risk Correlation Asset (% per year) (% per year) Exxon Boeing MG Exxon 11.2% 13.9% ----- 0.41 0.13 Boeing 9.1 16.5 0.41 ---- -0.22 GM 12.1 15.8 0.13 -0.22 ----

More information would help Question: How do we describe the relationship between two random variables? Covariance - Correlation

15.063 Summer 2003 18