I have a serious question for you Joe. How did you calculate the chances of 5 numbers appearing in 10 spins?

First, I would like to say how much I admire what you're doing Third. It's no mean achievement to make a living from roulette, especially playing systems online, and with an RNG too! Kudos to you. I hope you continue to find success because I'm drawing inspiration from it.

The calculation is a little tricky, but I will try to explain it; let me know if anything's unclear.

I'm assuming you're aware of the binomial distribution formula? it calculates the chances of the number of "successes" in a number of trials. So in this case I took the number of trials to be 8, not 10, because we want to calculate the probability of the cluster occurring in the minimum number of trials, which means taking the end of the sequence at the point where the cluster is confirmed to end (which happens on the 8th spin in your screenshot).

There are 5 of #28 in the sequence, but we don't calculate the chance of

exactly 5 #28's in 8 spins, rather we want the probability of 5

or more #28's. Why? this is where it can seem paradoxical; the chance of exactly 5 hits is actually lower than the chance of 5 or more, because in the latter case we are adding up the probabilities of exactly 5 hits, 6 hits, 7 hits, and 8 hits. This is just a consequence of the law in probability which says you that you add probabilities which are mutually exclusive.

Rather than calculate those separate chances and add them up, it's easier to use what's called a

** Cumulative Distribution Function** or CDF. The CDF is actually defined as the chance of

**less than or equal to** some outcome, which in this case would be the probability that you would get 0..5 hits of #28 in 8 spins. But we don't want that, we want the probability of

at least 5, and to get this we need to subtract the CDF from 1.

Why? because it has to be true (ie 100% chance) that either #28 hit 0 to 4 times inclusive, or it hit 5 times or more. ie, in terms of probability :

chance that it hit 0 to 4 times + chance that it hit at least 5 times = 1

But the chance that it hit 0 to 4 times is the CDF (using 4 as the parameter), so

CDF + chance that it hit at least 5 times = 1, and rearranging we have

Chance that it hit at least 5 times = 1 - CDF

Calculating the CDF is tedious but there are calculators online to do it. Unfortunately none of the ones I've seen give you the degree of precision needed for this calculation (because the probability is very small). I use a statistical software package called Gretl (free) which can do this very easily and a lot more besides.

Here is the output from computing the above calculation :

` p = 1 - cdf(b, 0.027, 8, 4)`

Generated scalar p = 7.50545e-07

But remember this is the probability of a particular number (in our case #28) hitting at least 5 times in 8 spins. We need the probability that

any number will do it. That means we have to multiply the probability by 37 (because there are 37 ways it can happen, one for each number).

So the final probability is 37 * 7.50545e-07 = 0.000027770, which is 1 in 36,010.

Ok so that was quite tough for those who don't know basic probability. Let me know if you want anything clarified.