Roulette WorksheetMath 511.4.97The purpose of this worksheet is to solidify the concepts of expected value and standard error, and tointroduce the normal curve as a tool for approximating probabilities that would be painful to computeotherwise (even with the help of a calculator). It also introduces some mathematical terms for concepts wehave already discussed in class. You do not need to worry about these terms. They are mentioned here sothat you will be able to consult other texts, and so that you will know what people are talking about whenthey use mathematical terms to describe things that you already know.Suppose you perform an experiment such as spinning a roulette wheel five times in a row and each timebetting on the same column. A roulette wheel has 38 pockets. One is numbered 0, another is numbered 00,and the rest are numbered from 1 through 36. The croupier spins the wheel, and throws a ball onto thewheel. The ball is equally likely to land in any one of the 38 pockets. Before it lands, bets can be placed onthe table (see diagram).Let X be some result of the experiment that you are interested in. In this case, say X is the number oftimes you win in five spins of the wheel. In other words, the variable X represents the number of times youwin by spinning the roulette wheel five times and each time betting on the same column.Other results you may be interested in include the amount of money you win in five spins, the numberof times the ball lands in a red pocket in five spins, the number of times the ball lands in the pocket labeled00 in five spins, and so on. For now we will concentrate on the variable X as described above. You may wishto choose a different definition for X later. A variable that gives the outcome of an experiment is called arandom variable (the mathematical definition is more technical, but this is the main idea).You may wonder why the experiment consists of five spins of the wheel rather than just one. I claimthat this is a natural way to view roulette. When you go to play roulette do you bet just once and thenwalk away, or do you play several times in a row? Repeating this experiment then simulates you comingback to the roulette table several times during the course of an evening. Or you could view repeating thisexperiment from the perspective of the casino. In this case repeating the experiment could represent manydifferent people playing roulette over the course of an evening.In the experiment we have described what are the possible values, x1,x2,x3,x4,x5, and x6, for X? What isthe probability that each of these values occurs? Let pibe the probability that the value xioccurs. Formallypi= P (X = xi). Compute p1,p2,p3,p4,p5, and p6. An assignment of probabilities pito the possibleoutcomes xiof an experiment is called a probability density function.1What is the box model for this experiment? In other words, if you were to translate this experiment intothe language of boxes and tickets, what would be the different values you would find on tickets? How manytickets of each kind would there be? Note that so far the book has been considering one spin of the wheel tobe the experiment, and they consider repeating the experiment some number of times. Here the experimentis spinning the wheel five times and you are repeating it once (for now).What is the expected value for X? If you were to repeat this experiment many times and each time computeX, the number of times you won, then take the average number of times you won from all these times, youwould probably get close to the expected value for X. We write “the expected value for X”asE[X]. Thereare two ways to compute E[X]. One is to take the average value of the tickets in the box. This is the methodyour book suggests. The other is to use the following formula:If X is a random variable with possible values x1,x2,···,xnand probability density function p1,p2,···,pn,then the expected value of X is E[X]=x1p1+x2p2+··· +xnpn.This definition is the one you will find in most books on probability. However I think your book’s definitionis the most intuitive.Compute E[X] according to the definition in your book.Compute E[X] according to the definition above.If you were to perform this experiment with an actual roulette wheel, the value you would get for X wouldprobably not be exactly E[X]. Why not?2If you were to perform this experiment you would probably get a value for X which would be somewhere nearE[X]. It would be E[X] plus or minus some amount of chance error. If you were to perform this experiment(spinning the roulette wheel five times) over and over again you would probably see that 68% of the timeyou would win E[X] times plus or minus SE dollars, where SE is the standard error of the experiment. 95%of the time you would win E[X] plus or minus two times SE times. 99% of the time you would win E[X]plus or minus three times SE times. What is SE for this experiment? If you were to repeat this experimentten times (for a total of fifty spins), what would be the standard error?To convince yourself that these theoretical numbers give realistic answers, you could either fly to Las Vegasand try the wheel yourself, or you could pull up data desk and have it simulate things for you. To dothis do Manip → Generate Random Numbers... and fill in: Generate 1 variable with 10 cases,#Bernoulli trials/ experiment = 5, and Prob(success) = .3 (note that .3 is approximately1238). Nextdo Plot → Histograms. Choose Plot Scale... from the hyperview menu and set the bar width to 1 (if it’snot set to 1 already). This will perform the experiment 10 times (for a total of fifty spins) and record thenumber of times you win for each experiment. Do this again for 10 experiments. Try increasing the numberof experiments to 30 ( do Generate 1 variable with 30 cases), then 100, then 1000, and finally 10,000.What do you notice?The Central Limit Theorem says that when drawing at random with replacement from a box (such asusing the box model for this roulette experiment), the probability histogram for the sum of the tickets (thetotal number of times you win at roulette) will follow the normal curve. This is true when the histogramis converted to standard units and the number of draws from the box (the number of repetitions of theexperiment) is relatively large.Choose one of your histograms where the number of trials is 1000. The area of each bar in the histogramgives the percentage of the time that the value it is above appeared as