Unit 7: Chance VariationThe (real) Law of AveragesIf we repeat an experiment more and more times, the fraction oftimes an event occurs will get closer to the probability of thatevent, but the difference(number of times event occurs)− (probability)(number of trials)is likely to go up.Repeat with a spreadsheetModeling games with boxes of tickets1. Roll one die. If you get a 5 or a 6,you win $2; otherwise you lose $1.You roll 30 times.26246-1302. Roll two dice. If you get a 7 or 11,you win $7; otherwise you lose $3.You roll 10 times.83672836-3103. Toss 3 coins. If you get 3 heads, youwin $9; otherwise you lose $2. Youtoss 40 times.18978-2404. Roulette, playing “splits” (2 numbersof 38 are winners). It pays 17 to 1,and you are betting $1 each time.You play 25 times.238173638-125EV and SE — Two of each!Suppose we know the box (model) of a game, and we make manydraws from the box with replacement (play the game many times).What should we expect to get for a sum of draws (Expected Valueof sum) and for the average of the draws (EV of avg)?And how much variability should we expect in the sum (StandardError of sum) and in the average (SE of avg)?Why the EV and SE are really an AV and SDGiven a box model, we could make a “superbox” of all the possiblesets of n draws from the original box.Then we could replace each ticket in the superbox by a ticket withits sum (so that some sums are probably more frequent thanothers).The EV of the sum is the AV of this new superbox.And the SE of the sum is the SD of this new superbox.The EV and SE of the average draws are just the EV and SE,respectively, of the sum divided by n.A simple, sorta general exampleSay the original box has two-fifths a’s and three-fifths b’s, where aand b are some numbers:.4 a.6 bThree draws is like one draw from the superbox(.4)(.4)(.4) a, a, a(.4)(.4)(.6) a, a, b (.4)(.6)(.4) a, b, a• • •(.6)(.6)(.6) b, b, bTaking the sum of the three draws is like one draw from the newsuperbox(.4)(.4)(.4) a + a + a(.4)(.4)(.6) a + a + b• • •(.6)(.6)(.6) b + b + bThe example IIBecause we don’t care what order we add, there are only four kindsof tickets in the second superbox:30(.4)3a + a + a31(.4)2(.6) a + a + b32(.4)(.6)2a + b + b33(.6)3b + b + bSo the average of the second superbox, i.e., the EV of the sum ofthree draws from the original box, isEV of sum =30(.4)3(3a) +31(.4)2(.6)(2a + b)+32(.4)(.6)2(a + 2b) +33(.6)3(3b)The example IIIThe coefficient of a in this mess is3(.4)30(.4)2+31(.4)(.6)23+32(.6)213= 3(.4)20(.4)2+21(.4)(.6) +22(.6)2= 3(.4)[.4 + .6]2= 3(.4)Similarly, the coefficient of b is 3(.6), soEV of sum = 3(.4a + .6b) = 3(AV of original box)The example IVSimilarly, the SD of the second superbox, i.e., the SE of the sum ofthree draws from the original box, isSE of sum =√3(SD of original box)The example IVSimilarly, the SD of the second superbox, i.e., the SE of the sum ofthree draws from the original box, isSE of sum =√3(SD of original box)Shortcut formula for SD of a 2-value boxSD = (larger − smaller)·p(fraction with larger)(fraction with smaller)Shortcut formula for SD of a 2-value boxSD = (larger − smaller)·p(fraction with larger)(fraction with smaller)Shortcut formula for SD of a 2-value boxSD = (larger − smaller)·p(fraction with larger)(fraction with smaller)Ex: Roll two dice. If you get a 7 or 11, you win $7; otherwise youlose $3.Avg = (8/36)7 + (28/36)(−3) = −7/9SD = (7 − (−3))p(8/36)(28/36)= (10/9)√14 ≈ 4.1683672836-3In general:n = number of draws (plays)EV for sum = (AV of box) ·nEV for avg = AV of boxSE for sum = (SD of box) ·√nSE for avg = (SD of box)/√nSE for sum gets larger as n gets larger, but only as√n.So SE for avg actually gets smaller as n gets larger.I don’t know why the authors decided to put the sum and avgformulas into different chapters.Also, they always compute the SE for avg as (SE for sum)/n —but it is the same as the one above. Honest!Example of EV and SERoll one die: If you get a 5 or 6, you win $2; otherwise you lose $1.You play 30 times.AV of game (box) =26(2) +46(−1) = 0SD of game (box) = (2 − (−1))/r26·46=√2EV of sum of 30 plays = 30(0) = 0EV of avg of 30 plays = 0SE of sum of 30 plays =√2√30 = 2√15SE of avg of 30 plays =√2/√30 = 1/√15Another exampleRoll 2 dice. If you get a 7, you win $4; if 11, win $8; otherwise,lose $2. Play 60 times.AV =636(4) +236(8) +2836(−2) = −49SD =r636(4 − (−49))2+236(8 − (−49))2+2836(−2 − (−49))2=29√185 ≈ 3.0Not the shortcut formula; it’s a 3-value box.EV of sum = −49(60) ≈ −26.7SE of sum =29√185√60 ≈ 23.4How EV’s and SE’s cha nge with nRoll two dice. If you get a 7 or 11, you win $7; otherwise you lose$3.AV =836(7) +2836(−3) = −79SD = (7 − (−3))r836·2836≈ 4.2n EV of sum EV of avg SE of sum SE of avg90 −70 −7/9 ≈ 39 ≈ .44900 −700 −7/9 ≈ 124 ≈ .149000 −7000 −7/9 ≈ 394 ≈ .04490000 −70000 −7/9 ≈ 1240 ≈ .014Normal approx with EV and SE in place of AV and SD (I)Again: Roll two dice. If you get a 7 or 11, you win $7; otherwiseyou lose $3. Play 90 times. What is the probability of winning atleast $5?Again, EV of sum = −70, SE of sum ≈ 39.So 5 in std units is z = (5 − (−70))/39 ≈ 1.9.P(win ≥ 5) = P(z ≥ 1.9) =100−942% = 3%Normal table (Area between −z and z )z Area(%) z Area(%) z Area(%) z Area(%)0.0 0.0 1.15 74.99 2.3 97.86 3.45 99.9440.05 3.99 1.2 76.99 2.35 98.12 3.5 99.9530.1 7.97 1.25 78.87 2.4 98.36 3.55 99.9610.15 11.92 1.3 80.64 2.45 98.57 3.6 99.9680.2 15.85 1.35 82.3 2.5 98.76 3.65 99.9740.25 19.74 1.4 83.85 2.55 98.92 3.7 99.9780.3 23.58 1.45 85.29 2.6 99.07 3.75 99.9820.35 27.37 1.5 86.64 2.65 99.2 3.8 99.9860.4 31.08 1.55 87.89 2.7 99.31 3.85 99.9880.45 34.73 1.6 89.04 2.75 99.4 3.9 99.990.5 38.29 1.65 90.11 2.8 99.49 3.95 99.9920.55 41.77 1.7 91.09 2.85 99.56 4 99.99370.6 45.15 1.75 91.99 2.9 99.63 4.05 99.99490.65 48.43 1.8 92.81 2.95 99.68 4.1 99.99590.7 51.61 1.85 93.57 3 99.73 4.15 99.99670.75 54.67 1.9 94.26 3.05 99.771 4.2 99.99730.8 57.63 1.95 94.88 3.1 99.806 4.25 99.99790.85 60.47 2 95.45 3.15 99.837 4.3 99.99830.9 63.19 2.05 95.96 3.2 99.863 4.35 99.99860.95 65.79 2.1 96.43 3.25 99.885 4.4 99.99891 68.27 2.15 96.84 3.3 99.903 4.45 99.99911.05 70.63 2.2 97.22 3.35 99.9191.1 72.87 2.25 97.56 3.4 99.933Normal approx with EV and SE in place of AV and SD (II)Roulette, playing splits