Probabilistic Models

Probabilistic Models

Probabilistic Models

The toy roulette at the left is a pale model of a real roulette wheel. Real roulette wheels are usually found in casinos, surrounded by glitter and glitz. But this toy captures the essentials of roulette. Both the toy and real roulette wheels have 38 slots, numbered 1 through 36, 0, and 00. Two of the slots are coloured green; 18 are coloured red and 18 are coloured black. Batters often bet on red. If they wager $1.00 on red then if the roulette ball lands in a red slot they win $1.00 but if it lands in either a green slot or a black slot they lose $1.00. Because there are 18 red slots out of a total of 38 slots the chances of winning this bet are 18/38 — considerably less than even. The casinos make up the rules and they make them up so that they make huge profits.Image result for Probabilistic Models in Operation Research diagram
Gambling games like roulette are good models for many phenomena involving chance — for example, investing in the stock market. It is easier to analyse games involving a roulette wheel than investments involving the stock market but the same ideas are involved. In this section, we will consider and compare two different strategies that a gambier might use playing roulette. The same kinds of strategies and considerations are involved with investments. The same tools that we develop here for roulette can be used by investors.
Suppose that you have $10.00 and that you want to win an additional $10.00. We will consider two different strategies:
 The Flamboyant Strategy: You stride purposefully up to the wheel with a devil-may-care smile on your face. You bet your entire fortune of $10.00 on one spin of the wheel. If the ball lands in a red slot then you win, pocket your winnings, and leave with $20.00 and a genuinely happy smile on your face. If the ball lands in a slot of a different colour then you smile bravely at everyone as if $10.00 is mere chickenfeed and leave with empty pockets and feeling gloomy. With the flamboyant strategy, your chances of winning are 18/38 or roughly 0.4737.
 The Timid Strategy: With this strategy, you approach the roulette table with obvious trepidation. After watching for a while and working up your courage, you bet $1.00. When the ball falls in a slot you either win or lose $1.00. Now you have either $9.00 or $11.00. You continue betting one dollar on each spin of the wheel until you either go broke or reach your goal of $20.00.
Before continuing pause and think about these two strategies. Which of the two do you think gives you the best chance of winning? or are your chances of winning the same whichever strategy you use?
One way to study the questions raised above is by trying the two strategies in real casinos, wagering your own real money. This approach has several advantages and several disadvantages. One advantage is that this approach is realistic. Real casinos are run by people who know how to make a profit. They are skilled at creating an atmosphere that is likely to encourage customers to bet and lose more than they might like. The lessons that you learn in a real casino are more likely to be real lessons than the ones you learn in a simulated casino like the one we use below. One disadvantage is that this approach can be very costly both in terms of money and time.
We take a different approach — using the CAS window to simulate playing with the second, or timid, strategy. We already know the chances of winning with the first, or flamboyant, strategy — 18/38, or roughly 0.4737.
Computer algebra systems like Maple, MathCad, Mathematica, or the CAS system in the TI-92 have a procedure that generates random numbers. For example, on the TI-92 the command rand(), produces a random number between zero and one. The screen below shows the results of executing this command seven times. Notice that it produced seven different random numbers.