PROBABILISTIC MODELS - Quantitative Techniques for management

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 colored green; 18 are colored red and 18 are colored black. Betters 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.

Gambling games like roulette are good models for many phenomena involving chance — for example, investing in the stock market. It is easier to analyze 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 gambler 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 genuine happy smile on your face. If the ball lands in a slot of a different color 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 randO, 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.

results of executing this command seven times

Using the random number generator in your CAS window, you can easily simulate one spin of a roulette with a procedure like the one shown below.

simulate one spin of a roulette

Your CAS window has a program that is built on this basic idea and will simulate playing roulette using the timid strategy. Use this program to answer the questions below.

  • Compare the timid strategy to the flamboyant strategy.
  • Consider an intermediate strategy — betting $2.00 on each spin of the wheel.
  • Consider another, intermediate strategy — betting $5.00 on each spin of the wheel.
  • Some people enjoy gambling. If you play the flamboyant strategy then you spin the wheel just once. On the average how often would you spin the wheel with each of the strategies above.
  • What conclusion can you draw from our work in this module regarding the advisability of diversifying your investments? Be careful. Your answer depends on your investment goals and your beliefs about whether stock prices are more likely to rise or to fall.

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