Critical Thinking Project 2

Daysha Serrano Professor Worden Math 130 Section B April 10th, 2023 Critical Thinking Project 2 Playing Roulette We all dream of winning big, becoming an instant millionaire; but how likely is that? Let's say we decide to pursue our goal of winning 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 wheel that is marked with numbered squares that are red, black, or green. Half of the numbers 1-36 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 of bets, but two of the most common are to bet on a single number (1-36) 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 1 in most casinos, which means we will be paid $36 for each $1 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: because all numbers occur only once and there are 38 #’s total 1 38 including 0 and 00. 2. Calculate the probability that we will lose: because there is one winning slot/number and 37 slots that will result in a loss 37 38 3. If we bet $8 on the winning number, how much money will we win? 36(8)= $288 or 280 subtracting what we put in originally 4. What is the expected value of a bet on a single number if we bet $1? or - $0.05 35( 1 38 ) + (− 1) 37 38 = 35 38 + −37 38 = −1 19

5. For a $5 bet, what is the expected value of a bet on a single number? 175( ) + (-5) = + - = or - $0.26 1 38 37 38 175 38 185 38 −5 19 6. What is the expected value of a bet on a single number if we bet $10? 350( ) + (-10) = + - = or -$0.53 1 38 37 38 175 19 185 19 −10 19 7. Do you see a pattern in the answers to the last three questions? Yes, as the amount used to bet increases, the expected value decreases, and as the amount used to bet doubled, the expected value doubled as well. These expected values show that for every bet placed on a number over a period of time, I have a chance of losing about $0.05 for every $1 betted. We decide that we can certainly increase our chances of winning if we bet on a color instead of a number. Roulette allows us to bet on either red or black and if the number is that color, we win. This bet pays even money in most casinos. This means that for each dollar we bet, we will win $1 for choosing the winning color. So, if we bet $5 and win, we would keep our $5 and win $5 more. If we lose, we lose whatever amount of money we bet, just as before. 8. What is the probability that we will win on a single spin if we bet on red? 18 38 9. What is the probability that we will lose on a single spin if we bet on red? 20 38 10. If we bet $60 on the winning color, how much money will we win? Is this more or less than we will win by betting $8 on our favorite number? Explain why. If we bet 60$ on the winning color, we will win back our $60 used to place the bet plus $60 more, winning a total of $60 on top of what you betted, doubling your money. This is less than what we will win if we betted $8 on our fav number ($288) because the payout is more ($36 won for every $1 used to place the bet)since you have less of a chance of winning because numbers don't repeat whereas there are 18 slots of red/black and 2 slots of green giving you more of a chance of winning but less payout. 11. What is the expected value of a bet on red if we bet $1? 1( ) + (-1) = + - = 18 38 20 38 9 19 10 19 −1 19 or -$ 0.05 12. For a $5 bet, what is the expected value of a bet on red? 5( ) + (-5) = + - = or -$ 0.26 18 38 20 38 45 19 50 19 −5 19 13. What is the expected value of a bet on red if we bet $10? 10( ) + (-10) = + - = or -$ 0.53 18 38 20 38 90 19 100 19 −10 19 14. Do you see a pattern in the answers to the last three questions? Yes, as the amount used to