P Chapter 5 Project Playing Roulette We all dream of winning big, becoming an in
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P Chapter 5 Project Playing Roulette We all dream of winning big, becoming an instant m onaire; but how likely is that? Let's say we decide to pursue our goal ofwinning big money by going to a casino and continually playing what we think will be an easy game: roulette. Can we expect to win big in the long run? Is one bet better than another? How do the casinos make so much money anyway? If we are betting against the casino, how do they make sure that they always win? This project will help you answer these questions. Let's begin with a lesson in roulette. Roulette is a casino game that involves spinning a ball on a heel that is marked with numbered squares that are red, black, or green. Half of the numbers l-36 are colored red and half are black and the numbers 0 and 00 are green. Each number occurs only once on the wheel. We can make many different types ofbets, but two of the most common are to bet on a single number (-36) or to bet on a color (either red or black). These will be the two bets we will consider in this project. After all players place their bets on the table, the wheel is spun and the ball tossed onto the wheel. The pocket in which the ball lands on the wheel determines the winning number and color. The ball can land on only one color and number at a time. We begin by placing a bet on a number between 1 and 36. This bet pays 36 to in most casinos, which means we will be paid $36 for each si we bet on the winning number. If we lose, we simply lose whatever amount of money we bet. 1. Calculate the probability that we will win on a single spin of the wheel 2. Calculate the probability that we will lose. 3. If we bet S8 on the winning number, how much money will we win? 4. What is the expected value of a bet on a single number if we bet SI? 5. For a $5 bet, what is the expected value of a bet on a single number? 6. What is the expected value of a bet on a single number if we bet SI0? 7. Do you see a pattem in the answers to the last three questions?Explanation / Answer
Given,
American Roulette game having numbers 1 to 36 and 0,00
colours are black and white for 1 to 36 and green for 0,00
1) total outcomes =1 to 36 and 0,00=38 outcomes
in a single spin, we can have one outcome
hence, probability of win=1/38=2.64%
2) probability of lose=1-probability of win
probability of lose =1-(1/38)
probability of lose =37/38 or 97.36%
3) As the casino pays 36 to 1
hence for a $8 bet, we can expect a return of 8*36=$288 upon winning
4) to calculate expected value, we need probabilities and possible outcomes
in this case,
outcome for win(x1) =36
probability for win(p1) =1/38
outcome for lose (x2) = -1
probability for lose (p2)=37/38
hence, expected value is
EV=x1p1+x2p2
EV=(36)*(1/38)+(-1)*(37/38)
EV=36/38-37/38
EV= -0.0264 or -2.64%
Hence Expected value for $1 if we bet on single number is -2.64%
5) As we are betting $5, we have
outcome for win(x1) =180
probability for win(p1) =1/38
outcome for lose (x2) = -5
probability for lose (p2)=37/38
EV=x1p1+x2p2
EV=(180)*(1/38)+(-5)*(37/38)
EV=180/38-185/38
EV= -0.1315 or -13.15%
Hence Expected value for $5 if we bet on single number is -13.15%
6) As we are betting $10, we have
outcome for win(x1) =360
probability for win(p1) =1/38
outcome for lose (x2) = -10
probability for lose (p2)=37/38
EV=x1p1+x2p2
EV=(360)*(1/38)+(-10)*(37/38)
EV=360/38-370/38
EV= -0.2631 or -26.31%
Hence Expected value for $10 if we bet on single number is -26.31%
7) Inferences
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