Viva Las Vegas! The Expected Winnings of Roulette

This post is a continuation of our study of probability by examining the popular casino game, roulette. If you missed it, check it out here.

Although the study on independence, and lack thereof, in roulette spins and bets is fascinating, personally we find the examination on the expected values of the main bets more elegant, and beautiful in their inerrant equivalence.

Quickly, here is a refresher on the color distributions across the numbered slots:
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• Slots 0, 00 are green;
• Slots 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are red;
• Slots 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 are black.

These numbers on the wheel are ordered in such a way that high and low numbers, as well as odd and even numbers, alternate in addition to red and black. The game then proceeds in the following way. First, the wheel is spun and a small ball is subsequently rolled into the groove, in the opposite direction as the motion of the wheel. Eventually the ball falls into one of the slots. If the slot matches a bet, then the player wins.

Now, expected value in this case is defined as the winning payout times the probability of winning minus the bet times the probability of losing. We know the probabilities of winning from the slot distribution above as well as the respective bet payouts below:

​Bet	​Payout	​Probability (W)	​Probability (L)
​Straight (Select one number)	35 : 1	1/38	37/38
Split (Select two numbers)	17 : 1	2/38	36/38
Street/Row (Select three numbers)	11 : 1	3/38	35/38
Square/Corner (Select four numbers)	8 : 1	4/38	34/38
Double Street/Row (Select six numbers)	5 : 1	6/38	32/38
Dozen/Column (Select twelve numbers)	2 : 1	12/38	26/38
High/Low  Red/Black     Odd/Even       (Select eighteen numbers)	1 : 1	18/38	20/38

For a simple bet of $1, we can calculate the expected value (or return) for each bet using the equation: (1 * Payout * P(Winning))  + (1 * (-1) * P(Losing)):

1. 1 * 35 * 1/38 - 1 * 37/38 = 35/38 - 37/38 = -2/38 = -1/19​
2. 1 * 17 * 2/38 - 1 * 36/38 = 34/38 - 36/38 = -2/38 = -1/19
3. 1 * 11 * 3/38 - 1 * 35/38 = 33/38 - 35/38 = -2/38 = -1/19
4. 1 * 8 * 4/38 - 1 * 34/38 = 32/38 - 34/38 = -2/38 = -1/19
5. 1 * 5 * 6/38 - 1 * 32/38 = 30/38 - 32/38 = -2/38 = -1/19
6. 1 * 2 * 12/38 - 1 * 26/38 = 24/38 - 26/38 = -2/38 = -1/19
7. 1 * 1 * 18/38 - 1 * 20/38 = 18/38 - 20/38 = -2/38 = -1/19

The updated chart then becomes:

​Bet	​Payout	​Probability (W)	​Probability (L)	​Expected Winnings
​Straight (Select one number)	35 : 1	1/38	37/38	-1/19
Split (Select two numbers)	17 : 1	2/38	36/38	-1/19
Street/Row (Select three numbers)	11 : 1	3/38	35/38	​-1/19
Square/Corner (Select four numbers)	8 : 1	4/38	34/38	-1/19
Double Street/Row (Select six numbers)	5 : 1	6/38	32/38	​-1/19
Dozen/Column (Select twelve numbers)	2 : 1	12/38	26/38	​-1/19
High/Low  Red/Black     Odd/Even       (Select eighteen numbers)	1 : 1	18/38	20/38	-1/19

The math does not lie, each bet is expected to return the same value, or in this case a loss, of -.052 cents per one dollar bet. This is the “house edge” that people reference when discussing gambling in casinos. In fact, if you calculate the expected for all casino games, you will find a similar negative value. The casino has to make money somehow!

In Part 3, we examine a popular gambling strategy and the probabilities of actually beating the house. Look out for that post in the upcoming week!

The SaberSmart Team