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{"sheet":"Computations","index":0,"n_rows":24,"n_cols":13,"rows_kept":24,"cut":false,"hidden":false,"merged":[[0,1,0,8],[5,7,5,10],[7,5,9,5],[12,4,12,5]]}
[null,"Computations"]
[]
[null,"Let  X  and  Y  have the joint distribution:"]
[null,"Note that X and Y are independent!"]
[]
[null,"Determine  E[X], E[Y], Var(X), Var(Y),",null,null,null,null,null,"X"]
[null,"StdDev(X), StdDev(Y), Cov(X,Y), Corr(X,Y),",null,null,null,null,null,2,3,5,15]
[null,"E[X+Y[, Var(X+Y), and StdDev(X+Y).",null,null,null,"Y",10,{"$f":"$L8*H$11","v":0.06},0.09,7.4999999999999997E-2,7.4999999999999997E-2,0.3]
[null,"(Of course, you can build any intermediate",null,null,null,null,20,8.0000000000000016E-2,0.12,0.1,0.1,0.4]
[null,"tables you find useful.)",null,null,null,null,30,0.06,0.09,7.4999999999999997E-2,7.4999999999999997E-2,0.3]
[null,null,null,null,null,null,null,0.2,0.3,0.25,0.25]
[]
[null,"answers",null,null,"formulas"]
[null,6.3,"E[X]",null,{"$f":"SUMPRODUCT(I15:I18,J15:J18)","v":6.3},null,null,null,"x","Pr(X=x)",null,"x+y","Pr(X+Y=x+y)"]
[null,20,"E[Y]",null,{"$f":"SUMPRODUCT(I21:I23,J21:J23)","v":20},null,null,null,2,0.2,null,12,{"$f":"H8","v":0.06}]
[null,26.310000000000002,"Var(X)",null,{"$f":"SUMPRODUCT(I15:I18^2,J15:J18)-E14^2","v":26.310000000000002},null,null,null,3,0.3,null,13,{"$f":"I8","v":0.09}]
[null,60,"Var(Y)",null,{"$f":"SUMPRODUCT(I21:I23^2,J21:J23)-E15^2","v":60},null,null,null,5,0.25,null,15,{"$f":"J8","v":7.4999999999999997E-2}]
[null,5.1293274412928644,"StdDev(X)",null,{"$f":"SQRT(E16)","v":5.1293274412928644},null,null,null,15,0.25,null,22,{"$f":"H9","v":8.0000000000000016E-2}]
[null,7.745966692414834,"StdDev(Y)",null,{"$f":"SQRT(E17)","v":7.745966692414834},null,null,null,null,null,null,23,{"$f":"I9","v":0.12}]
[null,26.3,"E[X+Y]",null,{"$f":"SUMPRODUCT(L15:L24,M15:M24)","v":26.3},null,null,null,"y","Pr(Y=y)",null,25,{"$f":"K8+J9","v":0.17499999999999999}]
[null,86.309999999999945,"Var(X+Y)",null,{"$f":"SUMPRODUCT(L15:L24^2,M15:M24)-SUMPRODUCT(L15:L24,M15:M24)^2","v":86.309999999999945},{"$f":"E16+E17","v":86.31},null,null,10,0.3,null,32,{"$f":"H10","v":0.06}]
[null,9.2903175403212099,"StdDev(X+Y)",null,{"$f":"SQRT(E21)","v":9.2903175403212099},null,null,null,20,0.4,null,33,{"$f":"I10","v":0.09}]
[null,null,null,null,null,null,null,null,30,0.3,null,35,{"$f":"K9+J10","v":0.17499999999999999}]
[null,null,null,null,null,null,null,null,null,null,null,45,{"$f":"K10","v":7.4999999999999997E-2}]

{"sheet":"Roulette","index":1,"n_rows":336,"n_cols":9,"rows_kept":336,"cut":false,"hidden":false,"merged":[[0,1,0,8],[28,1,28,2],[29,1,29,2],[21,1,21,2],[22,1,22,2],[27,1,27,2]]}
[null,"Gambling with the Casino's Money"]
[]
[null,"Many casinos offer promotional deals which give a guest, possibly along with room and"]
[null,"meal discounts, a quantity of special gambling chips. The special chips cannot be"]
[null,"cashed in for real money, but can be used to place bets at the gambling tables."]
[null,"Winnings are paid in regular chips, which can be cashed in. (The casinos do this in"]
[null,"order to \"force\" a guest to the gambling tables, and hope that, once the special chips"]
[null,"are used up, the gambler will continue to play.)"]
[]
[null,"Assume you've been given 300 $1 special chips, and you choose to use them at the"]
[null,"roulette table. Each time the wheel is spun, you bet one special chip on the \"1-12\""]
[null,"square (which has a probability of 12/38 of yielding you a payoff of $3, and 26/38 of"]
[null,"yielding you nothing.)"]
[]
[null,"What is the expected value of the \"real\" chips you'll be holding after 300 plays? What is"]
[null,"one standard-deviation's-worth of uncertainty in the value you'll be holding?"]
[]
[]
[null,"Your profit from any one play is a random variable  W  with",null,null,null,null,null,"w","Pr(W=w)"]
[null,"the distribution:",null,null,null,null,null,0,{"$f":"26/38","v":0.68421052631578949}]
[null,null,null,null,null,null,null,3,{"$f":"1-I20","v":0.31578947368421051}]
[null,"E[W] =",null,{"$f":"SUMPRODUCT(H20:H21,I20:I21)","v":0.94736842105263153}]
[null,"StdDev(W) =",null,{"$f":"SQRT(SUMPRODUCT(H20:H21^2,I20:I21)-D22^2)","v":1.3944885578412389}]
[]
[null,"Let  T  be the sum of 300 independent, identically-distributed"]
[null,"observations of  W.  Then"]
[]
[null,"number of plays",null,{"$f":"300","v":300}]
[null,"E[T] =",null,{"$f":"D28*D22","v":284.21052631578948},"(since T = W1 + W2 +…+ W300 is a sum of independent,"]
[null,"StdDev(T) =",null,{"$f":"SQRT(D28)*D23","v":24.153250327544772},"identically-distributed random variables)"]
[]
[]
[null,"The 300 plays are simulated, and the results tracked, below. (Each simulation"]
[null,"run involves 300 spins of the roulette wheel.)"]
[]
[null,{"$f":"SUM(H37:H336)","v":306},"total winnings",null,null,null,"play","profit"]
[null,null,null,null,null,null,1,{"$f":"IF(RAND()<=12/38,3,0)","v":0}]
[null,{"$f":"ADDRESS(ROW($B$36),COLUMN($B$36))","v":"$B$36"},"monitored cell",null,null,null,2,{"$f":"IF(RAND()<=12/38,3,0)","v":0}]
[null,284.22483,"mean",null,null,null,3,0]
[null,24.142547713470528,"sample standard deviation",null,null,null,4,0]
[null,186,"minimum",null,null,null,5,0]
[null,390,"maximum",null,null,null,6,0]
[null,100000,"number of simulation runs",null,null,null,7,0]
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[null,null,null,null,null,null,299,0]
[null,null,null,null,null,null,300,3]

{"sheet":"Gambling","index":2,"n_rows":41,"n_cols":7,"rows_kept":41,"cut":false,"hidden":false,"merged":[[0,1,0,8],[30,6,30,7],[31,6,31,7],[29,6,29,7]]}
[null,"Wagering \"Systems\""]
[]
[null,"Is there a winning system for playing a casino game, when the odds favor the house?"]
[null,"Sure - in theory."]
[]
[null,"If you bet on \"black\" at the roulette table, your chances of winning are 18/38, a bit less"]
[null,"than 50%. (When you win, you get your bet back and an equal additional amount.)"]
[]
[null,"So, bet $1. If you win, you're up a dollar. Go home."]
[null,"If you lose, bet $2 the next time. If you win, you're up  -$1 + $2 = $1. Go home."]
[null,"If you lose, bet $4 the next time. If you win, you're up  -$1 - $2 + $4 = $1. Go home."]
[null,"And so on, continually doubling your bet until you win. For sure, you'll eventually win,"]
[null,"and so you make $1 with certainty!"]
[]
[null,"What's wrong here?"]
[]
[]
[null,"The answer is simple: There's a positive probability that you'll lose anough times in a row"]
[null,"that you run out of money, and can't make the next doubled bet. (Or, you'll hit the"]
[null,"\"house limit,\" the maximum bet allowed by the casino.)"]
[]
[null,"Say that you start with $1,000,000. After 19 losses you'll only have $475,713 left, but"]
[null,"the system will require a bet of $524,288. You're screwed!"]
[]
[null,"Let's assume that you quit at this point. (So, your strategy is to start at $1, and"]
[null,"keep doubling on losses. Go home after your first win, or quit after 19 losses.)"]
[null,"What is the probability that you'll end up winning a dollar? What is your expected"]
[null,"profit on a given night?"]
[]
[null,{"$f":"1-(20/38)^19","v":0.99999494546065015},"Pr( P=$1)",null,null,"p","Pr(P=p)"]
[null,null,null,null,null,{"$f":"-(2^19-1)","v":-524287},{"$f":"(20/38)^19","v":5.0545393498238411E-6}]
[null,"Your odds of coming out ahead are",null,null,null,1,{"$f":"1-G31","v":0.99999494546065015}]
[null,"almost 200,000:1 . But"]
[]
[null,{"$f":"SUMPRODUCT(F31:F32,G31:G32)","v":-1.650034326640442},"E[P]"]
[]
[null,"The doubling system (and related so-called \"progressive\" gambling systems) is"]
[null,"seductive. You'll almost surely come out ahead. But the casinos love system players,"]
[null,"since - across hundreds of thousands of players, over thousands of nights - enough"]
[null,"system players lose BIG to more than compensate for the many who make a little"]
[null,"profit."]

{"sheet":"A,B,C","index":3,"n_rows":24,"n_cols":6,"rows_kept":24,"cut":false,"hidden":false,"merged":[[0,1,0,8]]}
[null,"Juggling a Portfolio"]
[]
[null,"Shares in companies A, B, and C have expected rates of return over the next"]
[null,"year of 10%, 11%, and 15%, respectively. One standard-deviation’s-worth of"]
[null,"uncertainty in what each stock will return is 6%, 8%, and 10%. The returns on"]
[null,"the stocks vary independently."]
[]
[null,"What is the expected value and standard deviation of the rate of return (over"]
[null,"the next year) on a portfolio consisting of equal dollar amounts invested in all"]
[null,"three stocks?"]
[]
[null,null,null,"A","B","C"]
[null,null,"E[ROR]",0.1,0.11,0.15]
[null,null,"StdDev(ROR)",0.06,0.08,0.1]
[]
[null,null,"allocation",{"$f":"1/3","v":0.33333333333333331},{"$f":"1/3","v":0.33333333333333331},{"$f":"1/3","v":0.33333333333333331}]
[]
[]
[null,"E[RORportfolio]",null,{"$f":"SUMPRODUCT(D16:F16,D13:F13)","v":0.12}]
[null,"StdDev(RORportfolio)",null,{"$f":"SQRT(SUMPRODUCT(D16:F16^2,D14:F14^2))","v":4.7140452079103175E-2}]
[]
[null,"Var( a∙RORA + b∙RORB + c∙RORC ) = a2∙Var(RORA) + b2∙Var(RORB) + c2∙Var(RORC)"]
[]
[null,"since the rates of return vary independently."]