Probability The Science of Uncertainty with Applications to Investments, Insurance,and Engineering Exl:Two coins are tossed and the outcome is observed.Before the coins are tossed we are given a choice ofthe following payoffs Payoff 1: Vin S1 for each head」 Lose $3 for getting twotails Payoff2: Win$I if the coins are differen Win $2 if both coins tum up tails Lose $3 if both coins turn up heads. Which payoff should we choose? Y1(HH)=2,Y1(HT)-=1,Y1(TH)=1,Y1(TT)=-3 And Y2(HH)=-3,Y2(HT)=1,Y2(TH)=1,Y2(TT)=2 (1 fy=-3,2 /1 fy=-3,2 py (y)= if y=1 Pr (y)= if y=1 0 otherwise 20 otherwise Py (y)=py (y)for all y To address these questions,let's consider a third payoff: Payoff 3:Win $3 for getting two heads. Win $I for getting one of each Lose $4 for getting two tails. fy=-4,3 p,()= if y=1 0 otherwise E[YIJ=E[Y2J=E[Y3] That is,all three payoffs have the same expected value. Probability: The Science of Uncertainty with Applications to Investments, Insurance, and Engineering Ex1: Two coins are tossed and the outcome is observed. Before the coins are tossed, we are given a choice of the following payoffs: Payoff 1: Win $1 for each head. Lose $3 for getting two tails Payoff 2: Win $1 if the coins are different. Win $2 if both coins turn up tails. Lose $3 if both coins turn up heads. Which payoff should we choose? S={HH,HT,TH,TT} Y1(HH)=2, Y1(HT)=1, Y1(TH)=1, Y1(TT)= -3 And Y2(HH)=-3, Y2(HT)=1, Y2(TH)=1, Y2(TT)= 2          = = − = otherwise if y if y p y Y 0 1 2 1 3,2 4 1 ( ) 1          = = − = otherwise if y if y p y Y 0 1 2 1 3,2 4 1 ( ) 2 p y p y for all y Y Y ( ) ( ) 1 2 = To address these questions, let’s consider a third payoff: Payoff 3: Win $3 for getting two heads. Win $1 for getting one of each. Lose $4 for getting two tails.          = = − = otherwise if y if y p y Y 0 1 2 1 4,3 4 1 ( ) 3 E[Y1]=E[Y2]=E[Y3] That is, all three payoffs have the same expected value