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{"sheet":"Game odds","index":0,"n_rows":115,"n_cols":15,"rows_kept":115,"cut":false,"hidden":false,"merged":[[63,3,63,9],[0,1,0,8]]}
[null,"Game Odds"]
[]
[null,"McDonald’s runs a promotion in which 500,000 stamps will be distributed. Three of those stamps read"]
[null,"“Instant Winner! $100,000!” If you buy eight food items with stamps attached, what are the odds that"]
[null,"you will receive (at least) one of these three stamps? Give your answer as “1 in x.”"]
[]
[null,500000,"stamps"]
[null,8,"collected"]
[null,3,"prize stamps"]
[]
[null,"This is an \"at least\" problem. Let's try, instead, finding the probability of receiving no winning stamps,"]
[null,"and then subtract that from 1."]
[]
[null,{"$f":"1-COMBIN(B7-B9,B8)/COMBIN(B7,B8)","v":4.7999328001258057E-5},"Pr(collecting a prize stamp)"]
[null,{"$f":"1/B14","v":20833.625003537345},"the odds of winning are 1 in …"]
[]
[]
[null,"In another promotion, McDonald’s prints 10,000 of each of 12 different types of stamps (each marked"]
[null,"with a Zodiac sign). A particular pair of stamps (Pisces and Aquarius) can be turned in for a free"]
[null,"Coca-Cola. If you collect a total of 8 stamps, what is the probability that you will be able to claim a"]
[null,"free Coke? "]
[]
[null,"Again, this is an \"at least\" problem (i.e., we're after the probability of winning at least one free Coke)."]
[null,"Let's flip it around: What's the probability of winning no Cokes? Well, to not win any Cokes, we must"]
[null,"fail to get any Pisces stamps, or fail to get any Aquarius stamps (or fail to get any of either type)."]
[null,"Let's break this \"or\" problem down, using Pr(A or B) = Pr(A) + Pr(B) - Pr(A and B), where A = \"we get"]
[null,"no Pisces stamps,\" and B = \"we get no Aquarius stamps.\""]
[]
[null,10000,"stamps of each type"]
[null,12,"types"]
[null,8,"stamps collected"]
[null,{"$f":"COMBIN((B30-1)*B29,8)/COMBIN(B30*B29,8)","v":0.49851967150616344},"Pr(A)"]
[null,{"$f":"B32","v":0.49851967150616344},"Pr(B)"]
[null,{"$f":"COMBIN((B30-2)*B29,8)/COMBIN(B30*B29,8)","v":0.23255718594200495},"Pr(A and B)"]
[null,{"$f":"B32+B33-B34","v":0.76448215707032197},"Pr(A or B)"]
[null,{"$f":"1-B35","v":0.23551784292967803},"Pr(get a free Coke)"]
[]
[]
[null,"McDonald’s cuts a deal with Michael Jordan. They will print a large (and equal) number of each of"]
[null,"twelve “action photo” trading cards showing highlights of Jordan’s basketball career. If you get a single"]
[null,"randomly-selected card each time you eat at McDonald’s, how many meals would you need to buy to"]
[null,"have at least a 50% chance of receiving all 12 cards? "]
[]
[null,12,"types of cards"]
[]
[null,"There's no quick way to work this problem. But we can break it into steps. There are, for example,"]
[null,"only two different ways to end up with 5 different cards after eating 8 meals: Have 5 different cards"]
[null,"after your first 7 meals, and then get a duplicate on your 8th meal - or have only four different cards"]
[null,"after your first 7 meals, and then get a new type of card on your 8th meal."]
[]
[null,"Therefore, Pr(5 types of cards after 8 meals) ="]
[null," Pr(5 types after 7 meals) * Pr(duplicate card on 8th meal | 5 types after 7 meals)"]
[null," Pr(4 types after 7 meals) * Pr(new type of card on 8th meal | 4 types after 7 meals)"]
[]
[null,"But Pr(duplicate card on 8th meal | 5 types after 7 meals) = 5/12, and"]
[null,"Pr(new type of card on 8th meal | 4 types after 7 meals) = 8/12"]
[]
[null,"And consequently, the 8th-meal probability can be calculated from the two 7th-meal probabilities. This"]
[null,"idea is used in the table below. (The first-row and first-column probabilities were calculated directly.)"]
[]
[null,"Scanning down the last column, we find that one would have to buy at least 35 meals in order to have"]
[null,"at least a 50% chance of collecting all 12 types of cards."]
[]
[null,null,null,"distinct cards collected"]
[null,null,null,1,2,3,4,5,6,7,8,9,10,11,12]
[null,null,1,1,0,0,0,0,0,0,0,0,0,0,0]
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[null,null,3,6.9444444444444441E-3,0.22916666666666663,0.76388888888888884,0,0,0,0,0,0,0,0,0]
[null,"number",4,5.7870370370370367E-4,4.4560185185185175E-2,0.38194444444444442,0.57291666666666663,0,0,0,0,0,0,0,0]
[null,"of",5,4.8225308641975306E-5,7.9571759259259248E-3,0.13261959876543208,0.47743055555555552,0.38194444444444448,0,0,0,0,0,0,0]
[null,"cards",6,4.0187757201646085E-6,1.3704025205761314E-3,3.9785879629629622E-2,0.25860821759259256,0.47743055555555558,0.2228009259259259,0,0,0,0,0,0]
[null,"collected",7,3.3489797668038406E-7,2.3208429783950612E-4,1.1088472007887516E-2,0.11604214891975306,0.3713348765432099,0.38990162037037035,0.11140046296296295,0,0,0,0,0]
[null,"(i.e., meals eaten)",8,2.7908164723365336E-8,3.8987706118541369E-5,2.9655215835048005E-3,4.6997070312499986E-2,0.23208429783950618,0.41156282150205759,0.25993441358024688,4.6416859567901224E-2,0,0,0,0]
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[null,null,41,6.8037783679663067E-44,8.2289167712212492E-31,4.5494925461499739E-23,1.3571275056584228E-17,2.040954406259166E-13,4.1875750413518821E-10,1.9761387733905545E-7,2.8850091963047132E-5,1.5419516627504246E-3,3.2619843059362844E-2,0.26874540632532606,0.6970637508277584]
[null,null,42,5.6698153066385893E-45,1.3714861285374984E-31,1.1373732051117999E-23,4.5237924733888385E-18,8.5048814444169647E-14,2.0949780774129257E-10,1.1548414053318328E-7,1.931573375758936E-5,1.1660804443838341E-3,2.7568690465156642E-2,0.25178659630810934,0.71945920135486896]
[null,null,43,4.7248460888654901E-46,2.2858102142296835E-32,2.8434331270700105E-24,1.5079393547619845E-18,3.5440021880052943E-14,1.0479851567907205E-10,6.7470497548227567E-8,1.2925274230281733E-5,8.8099891120707195E-4,2.3265428832059828E-2,0.235399161693293,0.74044141771387806]
[null,null,44,3.9373717407212424E-47,3.8096836903832387E-33,7.1085830081592104E-25,5.0264858416217343E-19,1.4767681076258569E-14,5.2419931185632723E-11,3.9410189494305615E-8,8.644962194166249E-6,6.6505760814873117E-4,1.9608107087851628E-2,0.21966013635752857,0.76005801452165245]
[null,null,45,3.281143117267702E-48,6.3494728173057588E-34,1.7771457837871667E-25,1.675500611977834E-19,6.1535355474971788E-15,2.6218580073444179E-11,2.3015487170604427E-8,5.7797290417334599E-6,5.0167486017627051E-4,1.6506353641913541E-2,0.20462314284237643,0.77836302588477979]
[null,null,46,2.7342859310564178E-49,1.0582454695509899E-34,4.4428645123801903E-26,5.5850153685194913E-20,2.5640848448312898E-15,1.3112879599124796E-11,1.3438810139555972E-8,3.8627424808100585E-6,3.7818272147944733E-4,1.3880713416638686E-2,0.19032227321249731,0.79541495445497779]
[null,null,47,2.2785716092136817E-50,1.7637424492516749E-35,1.1107161369137599E-26,1.8616751216548812E-20,1.068405918782161E-15,6.5579355157218835E-12,7.8458623545405461E-9,2.5807611580981875E-6,2.8492462193652218E-4,1.1661806860902102E-2,0.17677553601422896,0.81127514388935262]
[null,null,48,1.8988096743447343E-51,2.9395707487528125E-36,2.7767903569822533E-27,6.2055920692206308E-21,4.4518154399337807E-16,3.2795909946468982E-12,4.580032007906513E-9,1.7237765480465167E-6,2.1455372017175768E-4,9.789403539569216E-3,0.16398787582319357,0.82600643855720501]
[null,null,49,1.5823413952872788E-52,4.8992845812547044E-37,6.9419759169520559E-28,2.0685327723329776E-21,1.8549644705862035E-16,1.6400551865574453E-12,2.6733251334427893E-9,1.1510927120343054E-6,1.6148988231150043E-4,8.2114747130172857E-3,0.15195378676118895,0.83967209487580452]
[null,null,50,1.3186178294060656E-53,8.1654743020911883E-38,1.7354939833207511E-28,6.8951144475918625E-22,7.7291565296273374E-17,8.2013579953950685E-13,1.5602596887682392E-9,7.6850902682847134E-7,1.2150110930430342E-4,6.8832680647589461E-3,0.14065955031659277,0.85233491043923693]

{"sheet":"Traffic cohorts","index":1,"n_rows":59,"n_cols":13,"rows_kept":59,"cut":false,"hidden":false,"merged":[[46,4,47,4],[46,5,47,5],[44,1,44,12],[0,1,0,12],[24,1,24,3],[26,1,26,3],[28,1,28,3],[22,3,22,4],[8,3,8,12],[9,1,19,1]]}
[null,"Traffic Cohorts"]
[]
[null,null,"Eight cars pull out of a restaurant parking lot at roughly the same time. Due to on-going roadwork, they find"]
[null,null,"themselves on a long stretch of single-lane highway with no passing opportunities. Each driver has a different “ideal”"]
[null,null,"speed at which he would like to drive, and drives at that speed if unimpeded. Eventually the cars will be divided into"]
[null,null,"one or more “cohorts,” each cohort consisting of an unimpeded driver and his impeded followers. What is the"]
[null,null,"probability that the cars will break into 4 distinct cohorts?"]
[]
[null,null,null,"cars"]
[null,"cohorts",null,1,2,3,4,5,6,7,8,9,10]
[null,null,1,1,{"$f":"1/E10","v":0.5},{"$f":"1/F10","v":0.33333333333333331},0.25,0.2,0.16666666666666666,0.14285714285714285,0.125,0.1111111111111111,0.1]
[null,null,2,0,{"$f":"(1/E$10)*D11+(1-1/E$10)*D12","v":0.5},{"$f":"(1/F$10)*E11+(1-1/F$10)*E12","v":0.5},0.45833333333333331,0.41666666666666669,0.38055555555555559,0.35000000000000009,0.32410714285714293,0.30198412698412702,0.28289682539682548]
[null,null,3,0,{"$f":"(1/E$10)*D12+(1-1/E$10)*D13","v":0},0.16666666666666666,0.25,0.29166666666666669,0.3125,0.3222222222222223,0.32569444444444451,0.32551807760141094,0.32316468253968256]
[null,null,4,0,0,0,4.1666666666666664E-2,8.3333333333333343E-2,0.11805555555555558,0.14583333333333337,0.16788194444444449,0.18541666666666667,0.19942680776014113]
[null,null,5,0,0,0,0,8.3333333333333332E-3,2.0833333333333336E-2,3.4722222222222224E-2,4.8611111111111119E-2,6.1863425925925933E-2,7.4218750000000014E-2]
[null,null,6,0,0,0,0,0,1.3888888888888887E-3,4.1666666666666666E-3,7.9861111111111105E-3,1.2500000000000001E-2,1.7436342592592594E-2]
[null,null,7,0,0,0,0,0,0,1.9841269841269839E-4,6.9444444444444436E-4,1.5046296296296294E-3,2.604166666666667E-3]
[null,null,8,0,0,0,0,0,0,0,2.4801587301587298E-5,9.9206349206349179E-5,2.3974867724867719E-4]
[null,null,9,0,0,0,0,0,0,0,0,2.7557319223985884E-6,1.2400793650793649E-5]
[null,null,10,0,0,0,0,0,0,0,0,0,2.7557319223985883E-7]
[]
[]
[null,null,null,"Keys to the table:"]
[]
[null,"first column",null,null,"One car can only form one cohort."]
[]
[null,"first row",null,null,"n  cars form a single cohort precisely when the slowest car is in front."]
[]
[null,"rest of table",null,null,"n  cars form  k  cohorts only if (a) the fastest car is first, in a cohort of its own,"]
[null,null,null,null,"and the remaining  n-1  cars form  k-1  cohorts, or (b) the fastest car isn't first,"]
[null,null,null,null,"and the other  n-1  cars form  k cohorts."]
[]
[null,null,"For example,"]
[]
[null,null,null," Pr(8 cars form 4 cohorts)"]
[]
[null,null,null,"=","Pr(fastest car is first, and the other 7 form 3 cohorts) +"]
[null,null,null,null,"Pr(fastest car is not first, and the other 7 form 4 cohorts)"]
[]
[null,null,null,"=","1/8 * Pr(other 7 form 3 cohorts) + 7/8 * Pr(other 7 form 4 cohorts)"]
[]
[null,null,"Consequently, each probability in the table can be derived from the probabilities \"west\""]
[null,null,"and \"northwest\" of it."]
[]
[null,"Traffic Cohorts, using simulation"]
[]
[null,null,null,null,"speed (mph)","front of cohort"]
[null,null,null,"order",null,null,null,{"$f":"SUM(F49:F56)","v":2},"= number of cohorts"]
[null,null,null,1,{"$f":"30+40*RAND()","v":31.505588117021574},1,null,{"$f":"IF(H48=4,1,0)","v":0},"\"is the number of cohorts 4?\""]
[null,null,null,2,{"$f":"30+40*RAND()","v":32.423669122881954},{"$f":"IF(E50<MIN($E$49:E49),1,0)","v":0}]
[null,null,null,3,64.053413753682719,{"$f":"IF(E51<MIN($E$49:E50),1,0)","v":0},null,{"$f":"ADDRESS(ROW($H$49),COLUMN($H$49))","v":"$H$49"},"monitored cell"]
[null,null,null,4,49.842861799659246,{"$f":"IF(E52<MIN($E$49:E51),1,0)","v":0},null,0.16772999999999999,"mean"]
[null,null,null,5,53.338228127823633,{"$f":"IF(E53<MIN($E$49:E52),1,0)","v":0},null,0.37362821504863736,"sample standard deviation"]
[null,null,null,6,31.983808016413771,{"$f":"IF(E54<MIN($E$49:E53),1,0)","v":0},null,0,"minimum"]
[null,null,null,7,31.243750401459948,{"$f":"IF(E55<MIN($E$49:E54),1,0)","v":1},null,1,"maximum"]
[null,null,null,8,48.927307484478966,{"$f":"IF(E56<MIN($E$49:E55),1,0)","v":0},null,100000,"number of simulation runs"]
[]
[null,null,null,"So, we estimate the probability of 8 cars breaking into 4 cohorts to be",null,null,null,null,null,null,{"$f":"H52","v":0.16772999999999999}]
[null,null,null,"and the margin of error in the estimate (at the 95%-confidence level) is",null,null,null,null,null,null,{"$f":"1.96*H53/SQRT(H56)","v":2.3157716690075099E-3}]

{"sheet":"Race card","index":2,"n_rows":20,"n_cols":9,"rows_kept":20,"cut":false,"hidden":false,"merged":[[3,1,3,8],[4,1,4,2],[2,1,2,8],[0,1,0,4]]}
[null,"A Racing Card"]
[]
[null,"Calder    #9    06/20/2000    6 Furlongs    Dirt    Purse $7000    Maiden Claiming  $12500 "]
[null,"Exacta & Trifecta; Superfecta"]
[null,"P#  ",null,"  Horse Name","  ML","  Jockey","  Trainer","  Med","  Equip"]
[null,"pp 1","1  ","  Gate's Power","12:1","  Scocca Dante ","  O'connell Kathleen ","   ","   "]
[null,"pp 2","2  ","  Ms. Ritmo Latino","4:1","  Castellano Javi ","  Garoffalo Jose ","   ","   "]
[null,"pp 3","3  ","  Del Boca Vista","6:1","  Melton Randi ","  Hefner Paul ","   ","   "]
[null,"pp 4","4  ","  Ocean Slew","8:1","  Rivera; Ii J","  Julian William A","   ","  Bon"]
[null,"pp 5","5  ","  Nicole","15:1","  Portillo Dougla ","  Simone Victor ","   ","   "]
[null,"pp 6","6  ","  A Southern Grad","5:2","  Pitty Camilo D","  Rountree Cathy ","   ","   "]
[null,"pp 7","7  ","  Robyns Joy","10:1","  Bain Gary W","  Rolle Elliston C","   ","   "]
[null,"pp 8","8  ","  Her Richness","5:1","  Morales; Jr H","  Canet Julian ","   ","   "]
[null,"pp 9","9  ","  Honor The Bear","6:1","  Castellano Abel ","  Richards; Jr R","   ","   "]
[]
[null,"post position",null,null,"morning line",null,null,"medication","blinders on"]
[null,null,null,null,"(odds)"]
[]
[null,null,"The material on this and the following tabs is optional. It shows in detail how"]
[null,null,"parimutuel wagering works."]

{"sheet":"Parimutuel wagering","index":3,"n_rows":51,"n_cols":14,"rows_kept":51,"cut":false,"hidden":false,"merged":[[12,7,12,9],[0,1,0,5],[8,1,8,2],[8,3,8,4],[21,1,21,4],[12,4,12,5],[6,1,6,2],[6,3,6,4],[7,1,7,2],[7,3,7,4]]}
[null,"A Parimutuel-Wagering Example"]
[]
[null,"Calder Race Course (Miami)"]
[null,"2000-06-20"]
[null,"Race 9","4:18 PM post time"]
[null,null,null,null,null,1,"scratch",3,4,5,6,7,8,9]
[null,"Win Pool",null,52913,null,1682,"----",7941,2279,2042,15363,5339,15069,3198]
[null,"Place Pool",null,18489,null,747,"----",3402,1211,956,4735,1559,4823,1056]
[null,"Show Pool",null,6921,null,381,"----",1679,564,409,1430,500,1246,712]
[]
[null,"Finish: 3-6-9",0.18,"track takeout"]
[]
[null,null,"win",null,"place",null,null,"show"]
[null,null,"Horse 3",null,"Horse 3","Horse 6",null,"Horse 3","Horse 6","Horse 9"]
[null,"pool",{"$f":"D7","v":52913},null,{"$f":"D8","v":18489},null,null,{"$f":"D9","v":6921},null,null,null,"Note that cell comments"]
[null,"after takeout",{"$f":"C15*(1-$C$11)","v":43388.66},null,{"$f":"E15*(1-$C$11)","v":15160.980000000001},null,null,{"$f":"H15*(1-$C$11)","v":5675.22},null,null,null,"in column B explain the"]
[null,"after return",{"$f":"C16-H7","v":35447.660000000003},null,{"$f":"E16-H8-K8","v":7023.9800000000014},null,null,{"$f":"H16-H9-K9-N9","v":1854.2200000000003},null,null,null,"computations."]
[null,"shares",{"$f":"C17","v":35447.660000000003},null,{"$f":"E17/2","v":3511.9900000000007},null,null,{"$f":"H17/3","v":618.07333333333338}]
[null,"pre-breakage / $1 bet",{"$f":"1+C18/H7","v":5.463878604709735},null,{"$f":"1+E18/H8","v":2.0323309817754263},{"$f":"1+E18/K8","v":1.7417085533262937},null,{"$f":"1+H18/H9","v":1.3681199126464165},{"$f":"1+H18/K9","v":1.4322191142191143},{"$f":"1+H18/N9","v":1.8680805243445695}]
[null,"payoff / $2 bet",{"$f":"2*ROUNDDOWN(C19,1)","v":10.8},null,{"$f":"2*ROUNDDOWN(E19,1)","v":4},{"$f":"2*ROUNDDOWN(F19,1)","v":3.4},null,{"$f":"2*ROUNDDOWN(H19,1)","v":2.6},{"$f":"2*ROUNDDOWN(I19,1)","v":2.8},{"$f":"2*ROUNDDOWN(J19,1)","v":3.6}]
[]
[null,"Calder - Race 9 ",null,null,null,null,null,null,null,null,null,"Had Horse 6 (the favorite) won …"]
[null,"#","Win","Place","Show"]
[null,3,10.8,4,2.6,"announced after the race",null,null,null,null,null,"win"]
[null,6,".",3.4,2.8,null,null,null,null,null,null,"Horse 6"]
[null,9,".",".",3.6,null,null,null,null,null,null,{"$f":"D7","v":52913},"pool"]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"L26*(1-$C$11)","v":43388.66},"after takeout"]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"L27-K7","v":28025.660000000003},"after return"]
[null,null,null,{"$f":"SUM(D7:E9)","v":78323},null,"track receipts",null,null,null,null,null,{"$f":"L28","v":28025.660000000003},"shares"]
[null,null,null,{"$f":"D29*C11","v":14098.14},{"$f":"C11","v":0.18},"track takeout, pre-breakage",null,null,null,null,null,{"$f":"1+L29/K7","v":2.8242309444769904},"pre-breakage / $1 bet"]
[null,null,null,{"$f":"D29-D30-SUM(D35:D37)","v":1023.6599999999962},{"$f":"D31/D29","v":1.3069724091263054E-2},"breakage",null,null,null,null,null,{"$f":"2*ROUNDDOWN(L30,1)","v":5.6},"payoff / $2 bet"]
[null,null,null,{"$f":"D30+D31","v":15121.799999999996},{"$f":"E30+E31","v":0.19306972409126305},"total takeout"]
[]
[null,null,null,null,"paid out from …"]
[null,null,null,{"$f":"C20*H7/2","v":42881.4},"win pool"]
[null,null,null,{"$f":"E20*H8/2+F20*K8/2","v":14853.5},"place pool"]
[null,null,null,{"$f":"H20*H9/2+I20*K9/2+J20*N9/2","v":5466.3},"show pool"]
[]
[]
[null,"Consider the ninth race at Calder Race Course on June 20, 2000. (See the associated spreadsheet"]
[null,"for details.) If, after examining photographs taken at the finish line, the race judges had reversed"]
[null,"the finishing positions of the second- and third-place horses (so the order of finish was 3-9-6),"]
[null,"what would have been the payoff on a “place” bet on Horse 9? Should bettors holding “place”"]
[null,"bets on Horse 3 care about the outcome of the judges’ deliberations?"]
[]
[null,"Had the finish order been 6-3-9, the only change would have been for \"win\" bets. The place and"]
[null,"show payoffs would have been the same."]
[]
[null,"If you are going to collect on a place or show bet, you'd prefer for the other place/show horses to be"]
[null,"longshots. This would reduce the amounts \"returned\" from the pool, and therefore increase your"]
[null,"\"share.\""]

{"sheet":"Optimal bet","index":4,"n_rows":73,"n_cols":15,"rows_kept":73,"cut":false,"hidden":false,"merged":[[26,3,26,4],[0,1,0,4],[10,1,10,2],[10,3,10,4],[8,1,8,2],[8,3,8,4],[9,1,9,2],[9,3,9,4],[25,2,25,3]]}
[null,"Optimal Betting"]
[]
[null,"Suppose that, before the race, you had acquired information that led you to believe there was"]
[null,"a 20% chance that Horse 3 would win the race. Taking the tote-board (reproduced below) as"]
[null,"an accurate accounting of the bets placed by all other bettors, what would have been the"]
[null,"optimal amount for you to bet at the last minute (on Horse 3, to win)?"]
[]
[null,null,null,null,null,1,"scratch",3,4,5,6,7,8,9]
[null,"Win Pool",null,52913,null,1682,"----",7941,2279,2042,15363,5339,15069,3198]
[null,"Place Pool",null,18489,null,747,"----",3402,1211,956,4735,1559,4823,1056]
[null,"Show Pool",null,6921,null,381,"----",1679,564,409,1430,500,1246,712]
[null,"bettor assessments (to win)",null,null,null,{"$f":"F9/$D$9","v":3.1788029406762043E-2},null,{"$f":"H9/$D$9","v":0.15007654073668097},4.3070700961956421E-2,3.8591650445070208E-2,0.29034452780980097,0.10090147978757583,0.28478823729518266,6.0438833556970877E-2,{"$f":"SUM(F12:N12)","v":0.99999999999999989}]
[null,"your assessment",null,null,null,null,null,0.2]
[]
[null,null,null,478.41543750000005,"your bet (on Horse 3, to win)",null,null,null,"Determined by trial-and-error, using"]
[null,null,null,0.18,"track takeout",null,null,null,"values near $475."]
[null,null,null,{"$f":"D9+D15","v":53391.4154375},"win pool"]
[null,null,null,{"$f":"D17*(1-D16)","v":43780.960658750002},"after takeout"]
[null,null,null,{"$f":"D18-H9-D15","v":35361.545221250002},"after return"]
[null,null,null,{"$f":"D19","v":35361.545221250002},"share"]
[null,null,null,{"$f":"1+D20/(H9+D15)","v":5.2000000455791744},"pre-breakage / $1 bet"]
[null,null,null,{"$f":"2*ROUNDDOWN(D21,1)","v":10.4},"payoff / $2 bet"]
[null,null,null,{"$f":"-D15+D22*D15/2*H13","v":19.136617500000057},"expected profit"]
[]
[]
[null,null,"profit table"]
[null,null,"bet ","expected profit"]
[null,null,null,{"$f":"D23","v":19.136617500000057}]
[null,null,0,0]
[null,null,25,2]
[null,null,50,4]
[null,null,75,6]
[null,null,100,8]
[null,null,125,7.5]
[null,null,150,9]
[null,null,175,10.5]
[null,null,200,12]
[null,null,225,13.5]
[null,null,250,15]
[null,null,275,16.5]
[null,null,300,12]
[null,null,325,13]
[null,null,350,14]
[null,null,375,15]
[null,null,400,16]
[null,null,425,17]
[null,null,450,18]
[null,null,475,19]
[null,null,500,10]
[null,null,525,10.5]
[null,null,550,11]
[null,null,575,11.5]
[null,null,600,12]
[null,null,625,12.5]
[null,null,650,13]
[null,null,675,13.5]
[null,null,700,0]
[null,null,725,0]
[null,null,750,0]
[null,null,775,0]
[null,null,800,0]
[null,null,825,0]
[null,null,850,0]
[null,null,875,0]
[null,null,900,-18]
[null,null,925,-18.5]
[null,null,950,-19]
[null,null,975,-19.5]
[null,null,1000,-20]
[null,null,1025,-20.5]
[null,null,1050,-21]
[null,null,1075,-21.5]
[null,null,1100,-44]