Roulette

Roulette was invented by a French mathematician, Pascal.  It involves spinning a wheel and dropping a small ball along the edge.  When the wheel stops, the ball will have found its way into one of the slots.  The 38 slots contain the numbers 1 through 36 and the special numbers 0 and 0018 of the numbers are red, 18 are black and the 0 and 00 are green.

Definition of Probability

The probability of an event E, P(E), is defined by the fraction

Number of Ways E Can Occur
P(E)  =
Total Number of Possibilities

Example (The probability of Red)

The probability of the ball landing in Red is

18
P(Red)  =            =  9/19
38

The numerator corresponds to the 18 red slots and the 38 corresponds to the total number of slots in Roulette.

Below is a list of the probabilities of winning for each of the other bets in Roulette.

1 Number "Straight Up":  1/38

2 Numbers "Split" or "0/00":   2/38  =  1/19

3 Numbers "Street":  3/38

4 Numbers "Corner":  4/38  =  2/19

5 Numbers "First Five":  5/38

6 Numbers "Line":  6/38  =  3/19

12 Numbers "Dozen" and Columns:  12/38  =  6/19

18 Numbers "Red/Black," "High/Low", "Odd/Even":  18/38  =  9/19

Expected Value

It is essential to understand how good or bad a bet is.  We talk about how good the bet is by saying that if you make the same bet many many times, how much will you expect to win (or lose) per bet on average.  We will exhibit how to calculate the expected value with an example.

Example

Find the expected value for "Split" bet that pays 17 to 1 if you wager \$100.

Solution

First note that there are two possibilities.  The first is that one of your two numbers is hit.  In this case you win \$1700.  The second is that a different number is hit.  In this case you win -100 dollars.  To find the expected value, multiply the financial result by the probability of that happening.  Then add all of these products.

Since P(Win)  =  1/19 and a win results in 1700, we multiply

(1/19)(1700)  =  1700/19

To find the probability of losing, just subtract the probability of winning from 1 to get

P(Lose)  =  1 - P(Win)  =  1 - 1/19  =  18/19.

Losing results in winning -100, so we multiply

(18/19)(-100)  =  -1800/19

finally add these partial results to get

1700/19 + (-1800/19)  =  -100/19  =  -5.26

Now to interpret this number.  If you make the "Split" bet many times at \$100 per wager, you will average a loss of \$5.26 per wager.

Example

Find the expected value of the "Black" bet which pays even money.

Solution

We will proceed as in the last example, but with a shorter explanation.  Below is a probability distribution table, that is, a table that give the possible winnings and the probability of the winnings.  The third column is the product of the winnings and their probabilities

 x P(x) xP(x) 100 9/19 900/19 -100 10/19 -1000/19 Total -100/19  =  -5.26

The table below shows the expected values for each of the bets:

 Bet Payout Expected Value 1 Number "Straight Up" 35 to 1 -5.26 2 Numbers "Split" or "0/00" 17 to 1 -5.26 3 Numbers "Street" 11 to 1 -5.26 4 Numbers "Corner" 8 to 1 -5.26 5 Numbers "First Five" 6 to 1 -7.89 6 Numbers "Line" 5 to 1 -5.26 12 Numbers "Dozens" and "Columns" 2 to 1 -5.26 18 Numbers "Red/Black," "High/Low" or "Odd/Even" Even Money -5.26

Notice that the expected values are all the same, except for the "First Five" bet which has a worse expected value.

Standard Deviation

The standard deviation tells us how far from the mean the outcomes lie.  We will look at the standard deviation by example.

Consider the "Black" bet that was looked at earlier.  We will extend the table as follows

 x P(x) xP(x) (x - (-5.26))2 P(x)(x - (-5.26))2 100 9/19 900/19 11080 5248 -100 10/19 -1000/19 8976 4724 Total -100/19  =  -5.26 9972

The standard deviation is the square root of this number, that is the standard deviation is

s  =  100

Notice the Greek letter above.  It is read "sigma" and is the letter that represents the standard deviation.

Below is the table of all bets with the standard deviations shown

 Bet Payout Expected Value Standard Deviation 1 Number "Straight Up" 35 to 1 -5.26 576 2 Numbers "Split" or "0/00" 17 to 1 -5.26 402 3 Numbers "Street" 11 to 1 -5.26 324 4 Numbers "Corner" 8 to 1 -5.26 276 5 Numbers "First Five" 6 to 1 -7.89 94 6 Numbers "Line" 5 to 1 -5.26 219 12 Numbers "Dozens" and "Columns" 2 to 1 -5.26 139 18 Numbers "Red/Black," "High/Low" or "Odd/Even" Even Money -5.26 100

How can we use the standard deviation to understand gambling better?  Although there are many uses for the standard deviation, one important one is the following which we will demonstrate with some examples.

Consider Betting on "Black" 2500 times.  First calculate the square root of 2500 which is 50.  Next divide the standard deviation for "Black" by this square root, that is divide 100 by 50 to get 2.  Now add and subtract this result from the expected value:

-5.26 - 2  =  -7.26

-5.26 + 2  =  -3.26

The result is that if the game is played 2500 times then there is a 95% chance that the player will lose an average of between \$3.26 and \$7.26

It may seem like playing 2500 times will never happen; however, if you were to own a casino, then 2500 bets is not so many.  This is why the casino does not need to rely on luck.  Given enough wagers, the casino is practically guaranteed to win.  The more you play, the greater the chance you will end up behind.  Hence "quit while you're ahead" is a mathematically proven result.

Always decide how much you are willing to spend before you begin.  Never go beyond this number.  If you find yourself pulling out more cash to get back what you lost, then you have a gambling problem and should call the National Council on Problem Gambling at 1-800-522-4700.

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