P(x_n | x_(n-k) = x_n) ...

So I was in Vegas recently and was interested in the Roulette tables.  So they display a list of the most recent spins of the wheel.  So one thing I noticed is that almost all had at least a pair (many had more).  So what is the probability of this?  Could you use this to game the system (some people believe so).  The lists I saw had 14 elements on them.  So the probability of all of them being different is (assuming statistically independent runs):

Product of (38)!/((38-14)!(38^14)).  (There are only 38 possible outcomes.) Which is: 0.064394.

Sounds to me like a bad game. If you always ran the last 14 numbers (all on the board) and the payout on the field is only 30x, then after 2 spins you are in the hole even if you win.  Sounds like a bad deal.  So I did also notice that about 40% of the boards had a double spin within the last 5 elements.  So if you always played the last 5 and always one once in every 5 spins then you would make money.  So the probability of no repeats in 5 spins is: 0.76018.  So the probability (statistically) is 0.24.  So this means (from my informal data collection) that the probability of winning on the last 5 numbers is higher than it should be.  However this could be a data collection problem.

I will say this is similar to the birthday problem which is:  What is the probability of any pair of people having the same birthday in the room.  I have included a small table to help you out.  (Remember there is 365 days in 1 year.)  (This assumes all days are equally likely which is not true, and also people tend to group with other people like them which is dependent on birthday.  For instance if you a star hockey player and you are in a room with other hockey players the probabilities go WAY up.)

Two people : 0.27%
4 People : 1.64%
10 People : 11.70%
15 People : 25.29%
22 People : 50.71%  (Yes, you only need 22 people to have 50% probability of a double birthday.)

I would like to say however that the best thing to do whenever you are gambling is to walk away.  The house always wins.