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

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.

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 – 1st 12 numbers (1-12), 2nd 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

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• 1. Becky Anderson  |  November 17, 2009 at 6:39 pm

Did you play, or just realize that it makes no sense mathematically?

• 2. georgewoodbury  |  November 17, 2009 at 8:06 pm

I 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?