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Monotonic sequence of specific letter probability


Given length of character array (N), and number of each character appearing in this array (e.g. A is appearing nA times, B is appearing nB times etc, na+nB+..+nZ = N),
what is the probability of character (say, _c) appearing m (m <= n_c) times in a row in this array?

Let's say only characters are A and B.

nA = 5
nB = 3
N = 8
and suppose m = 3
and _c = 'A'
Situations of interest: AAA*****, *AAA****, **AAA*** etc.

I assume the probability is:
Do you agree?

Screenshot from 2016-02-06 19:58:28.png

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