## Roulette – A Nearly Perfect Game (Friday 11/13/09)

*November 14, 2009 at 10:39 am* *
2 comments *

*Sorry for the delay … Internet issues at the hotel …*

Since I am in Las Vegas at AMATYC 2009, I thought I’d take a look at the probabilities & expected values associated with roulette. Roulette is a game that features a spinning wheel with 38 numbered spaces (0, 00, 1-36). 18 of the numbers are red, 18 are black, and 0 & 00 are green. The dealer spins a small ball on the outside of the wheel, and the slot where the ball lands determines the winning number.

**Single Number Bet**

If you correctly predict the number that the ball lands in, you are paid 35 to 1. The expected value for this bet can be calculated by multiplying the probability of winning (1/38) by the return for winning (+$35), multiplying the probability of losing (37/38) by the return for losing (-$1), and totaling these products.

EV = (1/38)(+35) + (37/38)(-1) = -2/38 = -$0.053.

This expected value tells us that you can expect to lose $0.053 for every $1 you bet in this fashion.

**Two Adjacent Numbers**

The payout when you bet two adjacent numbers on the table is 17 to 1. The probability of winning is now 2/38 and the probability of losing is 36/38.

EV = (2/38)(+17) + (36/38)(-1) = -2/38 = -$0.053.

So, another bet with the same exact expected value.

**Other Bets**

Other bets you can make are:

- Green Bet – 2 green numbers 0 & 00
- Street Bet – 3 numbers in a horizontal row
- Corner Bet – 4 numbers forming a block
- Six Line – 2 adjacent horizontal rows
- Column Bet – 12 numbers in a vertical column
- Dozen Bet – 1
^{st}12 numbers (1-12), 2^{nd}12 numbers (13-24), last 12 numbers (25-36) - Odd or Even Bet – Number is odd (or even)
- Red or Black Bet – Number is red (or black)

The expected values for these bets are calculated in the same fashion.

Bet |
Payout |
P(Win) |
EV |

Green |
17 to 1 | 2/38 | -$0.053 |

Street |
11 to 1 | 3/38 | -$0.053 |

Corner |
8 to 1 | 4/38 | -$0.053 |

Six Line |
5 to 1 | 6/38 | -$0.053 |

Column, Dozen |
2 to 1 | 12/38 | -$0.053 |

Odd, Even, Black, Red |
1 to 1 | 18/38 | -$0.053 |

Notice that all of the expected values are the same, making roulette a perfect game. Almost.

**Top Line**

The Top Line bet is a 5 number bet: 0, 00, 1, 2, & 3. The payout for this bet is 6 to 1. Here is the expected value for this bet.

EV = (5/38)(+6) + (33/38)(-1) = -3/38 = -$0.079.

This bet has a lower expected value than all of the others, and should be avoided.

**Challenge**

What payout for the Top Line bet would make roulette a perfect game? Leave a comment with your answer.

**So – is there a good roulette strategy?**

Not really. If you play roulette you can expect to lose roughly 1 nickel for every dollar you bet. It’s not a game you play to get rich, you have to decide whether all of those nickels you are about to lose is worth the entertainment value of the game. And stay away from the top line!

*I am a math instructor at College of the Sequoias in **Visalia**, **CA**. If there are topics you’d like me to address in future Recreational math articles, send in your requests through the contact page on my web site. – George*

Entry filed under: Recreational Math. Tags: algebra, amatyc, black jack, Blackjack, casino, developmental math, education, gambling, george woodbury, Math, my math lab, MyMathLab, NADE, roulette, woodbury.

1.Becky Anderson | November 17, 2009 at 6:39 pmDid you play, or just realize that it makes no sense mathematically?

2.georgewoodbury | November 17, 2009 at 8:06 pmI stayed away. I do like to play games like blackjack or poker for social fun, but never really enjoyed roulette. In my mind, playing to have fun is the only reason, because it’s hard to end up winning. Did you play?