ReviewProbabilities in a Market Discussion Question A QuestionMultiplication RuleA QuestionThe Addition RuleThe Complement RuleRecall the Equally Likely RuleINDIRECT, IS ExpProbability From Fair PriceMath 5 CrewDepartment of MathematicsDartmouth CollegeProbability From Fair Price – p.1/10Review•Last time we looked at several bets:the Bush bet, the Lord of the RingsBet, and combinations of these bets.Probability From Fair Price – p.2/10Review•Last time we looked at several bets:the Bush bet, the Lord of the RingsBet, and combinations of these bets.•In the process, we learned the FFMPE(cX + dY ) = cE(X) + dE(Y ),Probability From Fair Price – p.2/10Review•Last time we looked at several bets:the Bush bet, the Lord of the RingsBet, and combinations of these bets.•In the process, we learned the FFMPE(cX + dY ) = cE(X) + dE(Y ),•and (provided X and Y areindependent!) the SFMPE(XY ) = E(X)E(Y )Probability From Fair Price – p.2/10Probabilities in a Market•If a random variable Z has theproperty that it is one if an event Uoccurs and zero otherwise,Probability From Fair Price – p.3/10Probabilities in a Market•If a random variable Z has theproperty that it is one if an event Uoccurs and zero otherwise,•then we say the Probability that Uoccurs is E(Z) and use the notationP (E) = E(Z).Probability From Fair Price – p.3/10Probabilities in a Market•If a random variable Z has theproperty that it is one if an event Uoccurs and zero otherwise,•then we say the Probability that Uoccurs is E(Z) and use the notationP (E) = E(Z).•Notice, from this view P (Bush is the next president) = E(X/100) = 0.6465, whilethe P (Lord of the Rings wins Best Picture) = 0.8385.Probability From Fair Price – p.3/10Discussion Question•Let Z be the bet which is one if atleast one pair of you mothers share abirthday (month and day) and zerootherwise. For what price would yoube willing to sell Z and for what pricewould you be willing to buy Z?Probability From Fair Price – p.4/10Discussion Question•Let Z be the bet which is one if atleast one pair of you mothers share abirthday (month and day) and zerootherwise. For what price would yoube willing to sell Z and for what pricewould you be willing to buy Z?•What do you feel would be Z’s FairPrice in an efficient market?Probability From Fair Price – p.4/10Discussion Question•Let Z be the bet which is one if atleast one pair of you mothers share abirthday (month and day) and zerootherwise. For what price would yoube willing to sell Z and for what pricewould you be willing to buy Z?•What do you feel would be Z’s FairPrice in an efficient market?•How about the bet W that at least 2pairs of your mothers share the samebirthday?Probability From Fair Price – p.4/10A Question•Let U be the event that Bush is thenext president and that Lord of theRings wins Best Picture. What isP (U)?Probability From Fair Price – p.5/10A Question•Let U be the event that Bush is thenext president and that Lord of theRings wins Best Picture. What isP (U)?•We need to find a bet which is 1 ifthey both win and zero otherwise.Notice, Z =X100Y100has this property.Probability From Fair Price – p.5/10A Question•Let U be the event that Bush is thenext president and that Lord of theRings wins Best Picture. What isP (U)?•We need to find a bet which is 1 ifthey both win and zero otherwise.Notice, Z =X100Y100has this property.•Hence using the FFMP and SFMP P (E) equalsE(Z) = E(X100Y100) =110000E(XY ) =5420.9010000= 0.542.Probability From Fair Price – p.5/10Multiplication Rule•If event U1and U2are independent,thenProbability From Fair Price – p.6/10Multiplication Rule•If event U1and U2are independent,then•P (U1and U2) = P (U1)P (U2)Probability From Fair Price – p.6/10Multiplication Rule•If event U1and U2are independent,then•P (U1and U2) = P (U1)P (U2)•For our UP (U) = (0.6465)(0.8385) = 0.5420Probability From Fair Price – p.6/10A Question•Let V be the event that Bush is thenext president OR Lord of the Ringswins Best Picture. What is P (V )?Probability From Fair Price – p.7/10A Question•Let V be the event that Bush is thenext president OR Lord of the Ringswins Best Picture. What is P (V )?•We need to find a bet which is 1 if V occurs and zero otherwise. The following bethas this propertyZ =X100+Y100−X100Y100Probability From Fair Price – p.7/10A Question•Let V be the event that Bush is thenext president OR Lord of the Ringswins Best Picture. What is P (V )?•We need to find a bet which is 1 if V occurs and zero otherwise. The following bethas this propertyZ =X100+Y100−X100Y100•Hence using the first and second fundamental mysteries P (V ) equalsE(Z) = E(X100+Y100−X100Y100)=E(X)100+E(Y )100−E(XY )10000= 0.6465 + 0.8385 − 0.5420 = 0.9430Probability From Fair Price – p.7/10The Addition Rule•For any events U1and U2P (U1or U2) = P (U1) + P (U2) − P (U1and U2)Probability From Fair Price – p.8/10The Addition Rule•For any events U1and U2P (U1or U2) = P (U1) + P (U2) − P (U1and U2)•and if U1and U2are independentP (U1or U2) = P (U1)+P (U2)−P (U1)P (U2)Probability From Fair Price – p.8/10The Addition Rule•For any events U1and U2P (U1or U2) = P (U1) + P (U2) − P (U1and U2)•and if U1and U2are independentP (U1or U2) = P (U1)+P (U2)−P (U1)P (U2)•For our VP (V ) = 0.6465 + 0.8385 − (0.6465)(0.8385) = 0.9430Probability From Fair Price – p.8/10The Complement Rule•For any event U , the event that U does notoccur is called is called U ’s complement anddenoted as Uc.Probability From Fair Price – p.9/10The Complement Rule•For any event U , the event that U does notoccur is called is called U ’s complement anddenoted as Uc.•We have the following Complement RuleP (Uc) = 1 − P (U)Probability From Fair Price – p.9/10The Complement Rule•For any event U , the event that U does notoccur is called is called U ’s complement anddenoted as Uc.•We have the following Complement RuleP (Uc) = 1 − P (U)•For example, the probability that George Bush fails to be the next president isP (Uc) = 1 − P (U) = 1 − 0.6465 = 0.3535Probability From Fair Price – p.9/10Recall the Equally Likely Rule•If a bunch of N outcomes are equally likely,then (by arbitrage) the market will assign anevent consisting of K of these outcomes theprobability...Probability From Fair Price – p.10/10Recall the Equally Likely Rule•If a bunch of N outcomes are equally likely,then (by arbitrage) the market
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