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### Roulette Questions

```
Date: 01/26/98 at 22:49:35
From: Humberto Castaneda
Subject: Probability/Statistics and gambling

Roulette Questions:

1. What is a safer bet: to bet on a number, or to bet on color?

2. Which bet maximizes your wins: betting on numbers, or betting on
colors?

Thank You.
```

```
Date: 01/27/98 at 20:32:13
From: Doctor Bill
Subject: Re: Probability/Statistics and gambling

Humberto,

As to the question about the safety of betting, you're always going to
lose in the long run, so no bet is "safe." As for which is the "better
bet," it turns out they are about the same.

If we are assuming you are playing roulette in America (in Europe the
roulette wheel is different) then there are 36 numbers, half red and
half black, and also a 0 and a 00, both green. (In Europe there is not
a 00.) The payoff for a bet on a number is 35-1 (notice that if it
were a "fair" game the payoff would be 38-1,because there are 38
different numbers, but that's how the house makes its money), and the
payoff for red or black is 1-1 (notice here again the game is not
quite fair since you will lose more often than you win because of the
two green numbers, but you are paid off as if you win and lose an
equal number of times).

In statistics and probability there is something called "mathematical
expectation," which is the AVERAGE amount you will expect to win on
any given play, if you play for a LONG time. (If the expectation is
negative you "win" a negative amount, or in other words you expect to
lose each time you play.) To find the expectation for a game you
multiply each payoff times the probability of getting that payoff, and
then add all of those numbers together. That total then is your
EXPECTION. You probably won't win (or lose) that particular amount on
any given play, but if you play for a long time you can "expect" to
win that amount, on the average, each time you play.

If we now look at the roulette game, suppose you bet \$1 on the number
5 each time you play. What are the payoffs? You can either win \$35 or
lose \$1. The probability that you win \$35 is 1/38 (only one winning
number out of the total of 38). The probability that you lose \$1 (that
is you "win" \$-1) is 37/38 (37 losing numbers out of the total of 38).
To find the expectation of this game do the following calculation:
35(1/38)-1(37/38). What do you get here? If it is positive you win and
if it is negative you lose.

Suppose you bet \$1 on RED each time you play. Here the payoffs are
that you win \$1 or lose \$1. The probability of winning is 18/38 and
the probability of losing is 20/38. (Do you see where these
probabilities come from?) To find the expectation do the calculation:
1(18/38)-1(20/38).

Notice that both of the calculations above came out negative, that is,
you're going to lose in the long run. In these two cases I think they
both came out the same, so it doesn't matter which way you bet your
money, you're going to lose it at the same rate. If they had come out
different, then the better bet would have been the bet with the
highest expectation.

-Doctor Bill,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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