Craps question ( from someone who has only played craps once)

Quote: mustangsally

How about the 14th Fibonacci number as a quick close answer?

I noticed that, if P(n) is the probability of not getting two (or more) wins in a row over n trials with individual win probability = p:
P(1) = 1
P(2) = 1 - p²
P(n) = (1-p) P(n-1) + p(1-p) P(n-2), since if the first trial is a loss, the probability is the same as for n-1 trials, and if the first trial is a win, the second needs to be a loss, at which point the probability is the same as for n-2 trials

I tried deducing a recursive formula, but I noticed some strange patterns:
the p⁰ term = 1
the p¹ term = 0
the p² term = 1 - n
the p³ term = n - 2
the p⁴ term = (n - 4) (n - 3) / 2
the p⁵ term = - (n - 4)²

I did a matrix-based recursive solution, and got:

P(n) = ( (1 - p²) (J^n-1 - K^n-1) + JK (J^n-2 + K^n-2) ) / (J + K)
where J = sqrt(1 + 2p - 3p²) / 2 + (1 - p) / 2
and K = sqrt(1 + 2p - 3p²) / 2 - (1 - p) / 2

For n = 12 and p = 244/495, this returns 0.0987983465, so it "looks accurate".

Just don't ask about runs without streaks of 3 or more, as the mathematics becomes more and more complicated when doing it this way.
For example, for runs without streaks of 5:
P(n) = (1-p) P(n-1) + p (1-p) P(n-2) + p² (1-p) P(n-3) + p³ (1-p) P(n-4) + p⁴ (1-p) P(n-5)
The only way I know to solve this requires calculating the eigenvalues for a 5 x 5 matrix, which I think results in solving a quintic equation (which has been proven not to have a general solution).

Quote: ThatDonGuy

The only way I know to solve this requires calculating the eigenvalues for a 5 x 5 matrix, which I think results in solving a quintic equation (which has been proven not to have a general solution).

Looks to me you have fun doing lots of math and that is so cool

Here is a fast video I just did
on how I would do the run of 5 in Excel

has no sound or music
should slow it down to see cell entries if one wants
and full size if you wear glasses as I do at the computer

Sally




I currently love the MonaLisa Twins



and If you have come this far
the Excel workbook is in my onedrive folder
link to it in me blog
file name: excel-markov-chain-easy-stuff-streaks.xlsb

here is a video on how to expand a sheet to do another run size.
I went from 7 to 8
wow, I only started with 5 and now have 2 thru 8, with colors
imagine
my gift to all