Biology: How to find the probability of randomly generating multiple, sequentially identical sets If I randomly generate a substring (example "ATGCAGC") with equal probability (1/X where X=4) for each digit with length (L) digits: What is the formula for finding the probability (P) of randomly generating that sequence (T) times, given a total string length (N)?
Example:
Given "ATGCAGC" string length L=7, number of possible characters X=4 with equal probability of being randomly generated 1/X.
In a case where N characters are generated, what is the probability that an exact substring with length L will occur T times?
If I have randomly, sequentially generated N=7000 characters, what is the probability that any exact substring length L=7 "ATGCAGC" will occur T=2,3,4... times? 
P is my dependent variable. L, T, N, X are independent. 
In terms of dice:
Example: If I sequentially roll a X=6 sided die N=7000 times: What is the P=probability I will roll the die sequentially the same (1,4,6,5,3,2,3) with sequence length L=7 for T=2 sequentially identical occurrences in the N=7000 sequential rolls of a single die? 
What is the probability in 7000 rolls I will have any 2 runs of 7 throws that have an exact sequential match? Example: (1,4,6,5,3,2,3 on rolls 201-207)  and (1,4,6,5,3,2,3) on rolls 5001-5007. It could be any number of (T) occurrences, on any roll numbers in (N) total die rolls.  
I am specifically solving for the probability, given any values for the independent variables. Overlapping or non-overlapping substrings or both are great. 
My question is related to (How many times will a consecutive sequence of throws randomly appear if I throw a four-sided die N times?)
 A: Very Crude method:
If there are X characters, there can be $X^L$ different number of ways you can generate a sequence with each X have $\frac{1}{X}$ probability.  Then each such sequence is equally likely.
Thus the probability that a sequence would appear out of the $X^L$ different ways is $\frac{1}{x^L}$.  
Then the total number of characters N that are generated will be split into $[\frac{N}{L}]$ blocks of those substrings.
The probability that a substring other than  the desired happens in any block is $(1-\frac{1}{X^L})$.
Now define the the random variable T, the  number of times such substring should occur where t = 0,1,... [N/L].
Take a case when T = 2, this substring should occur twice and the rest of the $[\frac{N}{L}] - T$ blocks should be other substrings of length L other than the desired for which we calculated the probability a couple of steps above. For N = 100 and L = 5, you will have 20 blocks with 5 strings each. You check the desired substring in each of these blocks. Then shift 1 character and check from 2-6,7-11,... and again shift 1 character from 3-8, 9-12,....  This you do it till you shift 5 times then you get to the original sequence 6-10,11-15,... Now you have in a way scanned all the N characters in 20 blocks five times to cover all consecutive appearances.  Once way you could accomodate is to multiply (the probability of finding substring) by L which in this case is 5.
Now we can safely assume these appearances of substrings to be a Binomial Distribution.
Thus
$$P( T = t) =\left({[\frac{N}{L}]\choose t} {(L*\frac{1}{x^L})}^t {(1-L*\frac{1}{X^L})}^{[\frac{N}{L}]-t}\right)$$
I hope this gives a real good approximation for the task that you have laid out.
Thanks
Satish
