STAT 321-01 2/1/08 Lab 4—Comparing Roulette Bets a.) Possible values of X: x1=-1 x2 =1 Probability P(X = x) 20/38 = 0.5263 18/38 = 0.4737 b.) E(X) = 05263.)47368(.1)52632(.1)(P)()(P)()(P22211−=+−==+===×∑iiixXxxXxxXx This means that in the long run, choosing colors at random will yield a net gain of -$0.05263 per play, so 100 plays would have a net cost of $5.26 and yield no profit on average. c.) Possible values of Y: y1=-1 y2 =35 Probability P(X = y) 37/38 = 0.97368 18/38 = 0.02632 d.) E(Y)= 05263.)02632(.1)97368(.1)(P)()(P)()(P22211−=+−==+===×∑iiiyXyyXyyXy The value is the same as randomly choosing colors results. e.) The net winnings are equal so the bets are technically equal in terms of net “gain,” but the thrill of play or opinion of the player may affect how one might choose to play. Playing numbers results in more significant gains at less frequent intervals, while playing colors results in minor gains and losses throughout yielding the same results in the long run as numbers. f.) I won 2 out of 5 times with a total gain = -1, and an average net gain of (-1-1-1+1+1)/5 = -.2. Others around me also had the same result. g.) Tally for Discrete Variables: C3 C3 Count -1 516 1 484 N= 1000 Descriptive Statistics: C3 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 C3 1000 0 -0.0320 0.0316 1.0000 -1.0000 -1.0000 -1.0000 1.0000 Variable Maximum C3 1.0000 h.) P(X = x1) = .526 ≈ Relative Frequency = 516/1000 = .516 P(X = x2) = .474 ≈ Relative Frequency = 484/1000 = .484 The relative frequency and the probability are very similar, varying only by .01 each. The mean is -0.032 according to Minitab, while the expected value is -.05263. This may be a result of not being run enough trials for the random results to tend towards a stable mean value. i.) parsum appears to add the previous total the next entry, showing a cumulative “net gain” after each event—“total gain so far.”10008006004002000100-10-20-30-40spinsnet-colorScatterplot of net-color vs spins 100080060040020001.000.750.500.250.00-0.25-0.50spinsavg-colorScatterplot of avg-color vs spins The total net winnings varies dramatically over time and does not appear to converge to any constant number in this interval, although it should theoretically approach -.05263(1000)= -$52.63; the graph shows that this value may be reached soon, although more trials may be necessary to attain more accurate results. The average appears to approach -.03, nearly the theoretical -0.05263. In this series of events, my results were “above average” since did not lose as much as I theoretically should have. j.) Tally for Discrete Variables: C13 C13 Count -1 983 35 17 N= 1000 Descriptive Statistics: C13 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 C13 1000 0 -0.388 0.147 4.656 -1.000 -1.000 -1.000 -1.000 Variable Maximum C13 35.000 P(X = y1) = .97368 ≈ Relative Frequency = 983/1000 = .983 P(X = y2) = .02632 ≈ Relative Frequency = 17/1000 = .017 The relative frequencies are each within about .009 from the theoretical probabilities. While they seem fairly close, when many trials are performed, this variation can have dramatic effects on the cumulative results. The mean is -.0388 according to Minitab, while the expected value is -.05263. This difference may be a result of not being run enough trials for the random results to tend towards a stable mean value that might occur after many more trials. 100080060040020000-100-200-300-400num spinsnet-numbersScatterplot of net-numbers vs num spins 10008006004002000-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9-1.0num spinsavg-numbersScatterplot of avg-numbers vs num spins The total net winnings decreases at a constant rate until a win boosts the winnings by $35 at random intervals with a theoretical probability of .02632 of occurring. Unlike thecolors net winnings, this case never broke even and only lost money, significantly more than the colors case—a difference of $356: | (net-colors)-(net-numbers)|=|(-32)-(-388)|=$356 The average does not appear to approach a limit because the big wins create erratic behavior in the graph. The general trend appears to be upward, perhaps converging to the theoretical -.05263. Despite that possibility, it is strange that the resulting mean is -.388; a whole order of magnitude off! I would attribute this result to the fact that the relative frequency of wins is lower than theoretically predicted and the effect of such a large difference between losses and wins that can significantly alter the net gain. k.) 100080060040020001.00.50.0-0.5-1.0SpinsMeanAvg-ColorAvg-NumbersVariableScatterplot of avg-color vs spins, avg-numbers vs num spins Both plots are similar in that they are nearly always negative values. Otherwise they are fairly different: color follows smooth forms while numbers is rough and erratic. Also, numbers is always more negative than colors in this case. Colors tends toward a stable value on the long run, however, numbers looks to level off slightly, but is still changing rapidly even after 1000 spins—a result of the large affects of the $35 win and its small probability. l.) 9972.)5263(.))05263.(1()4737(.))05263.(1()(σ222=−−−+−−== XV dollars2 9986.9972.)()(SD === XVX dollars Units are in dollars2 and dollars since these are the variance and standard deviation of the net gain, which is measured in dollars. m.) 2157.33)97368(.)1()02632(.35)(222=−+=YE 2129.33)05263.()2157.33()(2=−−=YV dollars2 7631.52129.33)(SD ==Y dollars n.) The standard deviation is larger for playing numbers. This makes sense because the difference between 1 and -1 (colors) is much smaller than 35 and -1 (numbers), therefore the difference between the expected value and the possible results will be much larger in the numbers games since the spread between the possible values is much larger—resulting in a larger standard deviation. o.) Standard Deviation of C3 (colors)Standard deviation of C3 = 0.999988 Standard Deviation of C13 (numbers) Standard deviation of C13 = 4.65609 The ‘colors’ theoretical and empirical standard deviation are very similar, which makes sense because the experimental mean was similar to the theoretical one as well. ‘Numbers’ did not return a very close empirical and theoretical standard deviation, varying by more than 1. I’ll