Winning Roulette

An American roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. This is called "coming in". Players may bet on which number the ball will land on by placing chips on a felt layout. Although the numbers on the wheel itself are in no particular order, the numbers on the felt layout appear in numerical order.


The amount of money a player wins depends on the type of bet placed.

1. If think the ball will land on a particular color, you can bet on red or black. If you are right, you will be paid $2 for each dollar you bet.

2. If you think the ball will land on one of the twelve numbers in a long rows of numbers on the layout, you may bet on one of these columns. If you are right, you will be paid $3 for each dollar you bet.

3. If you think the ball will land on one of three numbers laid out in a row across the layout (like 16, 17, or 18), you can bet on one of these "streets". If you are right, you will be paid $12 for each dollar you bet.

4. If you want to bet that one of two numbers will come up, and those numbers share a common border on the layout, you can make what is known as a "split bet". If the ball lands on either of these two numbers, you will be paid $18 for each dollar you bet.

5. If you want to bet that the ball will land on a certain number, you may bet on any one of the 38 numbers (including 0 and 00). If the ball lands on your number, you will be paid $36 for each dollar you bet.

Question 1: What is the probability of winning with each of the above bets?

Question 2: What is the expected value each play, taking into account the $1 cost of each play? Interpret your results.

Question 3: The casino essential bets will all of its customers. Assume 100,000 individual bets are made on blacks or reds in a week. How much money can the casino expect to win or lose during the week?

Question 4: In Europe, roulette wheels have 18 red and 18 black slots, but only one zero slot. What effect does this have on a players probability of winning and on the expected value of each play?


The Discrete Mathematics Project