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Roulette Probability

Date: 12/07/97 at 15:50:36
From: karol
Subject: Probability of roulette

Dear Doctor,

I'm a 18-year-old student in Toronto who has an ISP on the probability 
of winning in roulette. We were supposed to choose one aspect of the 
game so I chose the probability of winning a certain amount of money 
using only the even money bets and the two to one bets. This really is 
not my favorite subject and I can use all the help I can get.

Thank you, 

Date: 12/09/97 at 20:34:53
From: Doctor Tom
Subject: Re: Probability of roulette

Hi Tom,

In most of the world, roulette wheels have 36 numbers (1 through 36)
and a 0 and 00 ("double zero"), so there are 38 outcomes. This is
certainly true in the United States, and I'm pretty sure it's true
of Canadian wheels as well. In the casino at Monte Carlo, however,
there are the same numbers, 1 through 36, and only a single 0.

I'll work out the results for the US-type wheel.

Suppose you bet on black. 18 of the 36 numbers are red and the other 
18 are black. The 0 and 00 are green. Thus, in the long term, all the 
outcomes are equally likely, so of the 38 possible results, 18 will be 
favorable. On average, you will win 18 times out of 38, or your 
probability of winning is 18/38 = 9/19 - slightly less than half.

A typical two to one bet would be to bet on a group of 12 numbers
(there's a way to do this - bet on a column of numbers). Thus you will 
win 12 times out of the 38 possible, for a probability of winning of 

You can work out any other probability the same way. Just figure out 
how many of the results represent a win, and divide that by 38.

Now what's interesting is your "expected return". That turns out to be 
the same, no matter what bet you make.

For example, if you're betting on black, and you bet one dollar every 
time, on average, you'll win 18 times and lose 20. So for the 18 
wins, you'll get back the 18 dollars you bet plus 18 dollars you won. 
After 38 bets, on average, you will have bet 38 and gotten back 36 - 
about a 5% loss. If you make the two to one bets, you'll win 12 times 
out of 38. On each win, besides getting back your original bet, you 
get 2 dollars extra, so for each of the 12 wins, on average, you get 
3 dollars back for a grand total of 36 back for the 38 you bet.  It's 
the same payoff - 36 dollars back for every 38 you bet.

If you just bet on a single number and you win, you get 35 additional
dollars back, which is 36 total back if you include the original bet.  
Of course this only happens once in 38 times, so again, you get about 
36 back, on average, for every 38 you bet.

Hence every bet on the roulette wheel, on average, loses you about
5 percent. In fact, the loss, 2/38 = 5.2631579 percent.  This is 
called the "house advantage".

You can do the same calculations for the wheels in Monte Carlo - the 
payoffs are the same, but there are only 37 possibilities. See if you 
can figure out why the house advantage at Monte Carlo is only 2.7027 

-Doctor Tom,  The Math Forum
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Associated Topics:
High School Probability

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