**The Discrete Uniform Distribution**

This describes any set of outcomes having equal probabilities. The classic example is a dice throw; all outcomes have the same probability of 1/6. So of course it applies to roulette numbers too since each will hit with a probability of 1/37. Strictly speaking it doesn't apply to other bets on the table, (EC, dozs, lines, quads etc) because although each EC, doz, line etc has the same probability as any other in the same group, none of them includes the zero, so the sum of the probabilities doesn't quite add up to 1, as it should. However to a good approximation the DUD can be used on them too.

To use the DUD, each member of the group is coded with numbers starting at 1 and ending at n and the probability of a win is 1/n (eg. for roulette n = 37). So to code the singles code #0 = 1, #1 = 2, #3 = 4, up to #36 = 37.

To find the z-score of a DUD for some sample of outcomes, you need

*The average "score" of the sample, X

*The expectation, E

*The standard deviation STD

Then you plug those numbers into the formula for z, which is : Z = ( X - E ) / STD

Where E = ( n + 1 ) / 2

and STD = sqrt[ (n^2 - 1) / 12 ]

To find X, you need to get the average of the sample which involves adding up all the codes and dividing by the sample size.

Here's an example using lines (double streets).

First code each DS as 1-6 = 1, 7-12 = 2, 13-18 = 3, 19-24 = 4, 25-30 = 5, and 31-36 = 6

After 7 spins you have this sequence of 7 numbers which represent the DS's which have hit :

2,4,4,1,2,1,4

step 1 : To find X, add up all the numbers in the sequence and divide by 7:

(2 + 4 + 4 + 1 + 2 + 1 + 4 ) / 7 = 2.571

step 2 : calculate the expectation. There are 6 possible outcomes (ignoring zero) so n = 6 and

E = (6 + 1) / 2 = 3.5

step 3 : calculate the standard deviation. This is sqrt[( 6^2 - 1) / 12] = sqrt[ 35 / 12 ] = 1.708.

step 4 : calculate the z-score. We now have X, E, and STD so Z = ( 2.571 - 3.5 ) / 1.708 = -0.544

The z-score tells you how "skewed" the distribution is, so in this example the skew is towards the low numbers because z is negative. A score close to zero means little or no skew in the outcomes.

Next up, the Negative Binomial Distribution.