[BruceZ]

I have PMed irchans about this, and he agrees with my objections.

This method has a problem similar to the one in my solution.


[ QUOTE ]

The Probability p[n] that the nth window will have exactly 14 hands is 0.0014279740790007608. Let g[n] be the probability that the nth window has exactly 14 hands and that no previous window had 14 hands. Let r[n,i] be the probability that the nth window will have exactly 14 hands given that the ith window had exactly 14 hands. Note that r[n,i] = r[n-i] because it only depends on the difference between n and i. Then

p[n] = Sum[ g[j] * r[n-j] , {j, 1, n}].

[/ QUOTE ]

For this formula to be correct, r[n,i] must be the probability that the nth window will have exactly 14 hands given that the ith window had exactly 14 hands, and given that no window previous to i had 14 hands. This latter condition will change the probability.

Further, it is not true that r[n,i] only depends only on n-i, because when the windows overlap, small values for the starting position of the first window i place different constraints on the window starting at n than larger values of i, since there would be more potential windows with 14 pairs that could start before a large value of i, and the hands in the window starting at n would be constrained to preclude this.


[ QUOTE ]
Now we just need the sequence r[n] to compute g[n]. If n<100, r[n] is the sum over k=0,...,14 of the probability that the first n hands of a window has k pairs times the probability that the next n hands after the window have k pairs. (If n>=100, r[n] = 0.0014279740790007608.)

Code:


r[n] = Sum[ C[n, i]
*(1/17^i)*(16/17)^(n - i) * prfirst[n, i],
{i, 0, Min[n, 14]}]



The probability that the first n hands of a 100 hand window have i pairs is

Code:


prfirst[i, j] = If[ j < 14 - (100 - i),
0,
C[i, j]*C[100 - i, 14 - j] / C[100, 14]]

[/ QUOTE ]

Because of the conditional issue discussed above, all C[100,14] do not have the same probability when the window starting at n overlaps with the first window. In this case, arrangements which put more pairs at lower values of n would be less likely, since these arrangements are constrained to preclude a window of 14 starting before the first window. The C[i, j]*C[100 - i, 14 - j] would be similarly constrained.