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Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and let $B$ denote the event of drawing a blue ball. I'm conducting $r+b$ trials of this experiment. So, there are a total of $r+b \choose r$ equally likely possibilities.

Each of the $r+b \choose r$ possibilities can be considered as a string of $R$s and $B$s. Each such string has $r+b-2$ substrings (strings formed from the original string from consecutive $R$s or $B$s) of length $3$.

For example, if $r = 7$ and $b = 3$, then $RRBBRRBRRR$ (which may be denoted by $R_2B_2R_2B_1R_3$) is one of the $10 \choose 3$ possibilities. It has $8$ substrings of length $3$, which are $RRB$, $RBB$, $BBR$, $BRR$, $RRB$, $RBR$, $BRR$ and $RRR$.

Now, given the premise of $r$ red balls and $b$ blue balls, after conducting $r+b$ trials, I'm interested in finding the probability distribution of the substrings of length $3$ in the resulting string. How do I approach this problem?

Motivation:- The motivation behind this problem directly arises from a reaction problem in polymer chemistry involving two types of molecules ($R$ and $B$), where the aim is to find the probability of an $R$ (or alternately, a $B$) being flanked on both sides by $R$s or $B$s.

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  • $\begingroup$ One way to do this is with the transfer matrix method. (See, e.g., chapter 4 of Richard Stanley's Enumerative Combinatorics, volume 1.) The basic idea is that as you draw the balls, you keep track of the colors of the last two balls drawn. You can represent the possible transitions as edges in a directed graph with weights that keep track of the number of R's, B's, and triples of each type, so that the numbers you want can be obtained by extracting coefficients from powers of a 4 by 4 matrix, and from this matrix you can get a rational generating function for the numbers. $\endgroup$
    – Ira Gessel
    Commented Jan 20, 2015 at 17:43
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    $\begingroup$ Do you actually need the distribution? I'm asking since it's easy to calculate the expectation, variance and other moments and also asymptotically there's going to be a central limit theorem. $\endgroup$
    – Ori Gurel-Gurevich
    Commented Jan 20, 2015 at 20:52
  • $\begingroup$ @IraGessel: Thank you for that reference. I shall read up and get back to you. $\endgroup$
    – Train Heartnet
    Commented Jan 21, 2015 at 2:07
  • $\begingroup$ @OriGurel-Gurevich: Ideally, I am looking for the distribution, as that information is vital for the related chemistry problem. But I'm open to finding any and all info that can shed light on this problem. So could you elaborate on your answer of calculating the expected value, variance and the other easier calculated results? $\endgroup$
    – Train Heartnet
    Commented Jan 21, 2015 at 2:13
  • $\begingroup$ I'll write an answer later, although calculating the expectation and variance is standard and so more fitting at math.stackoverflow. What values of $r$ and $b$ are you looking at? $\endgroup$
    – Ori Gurel-Gurevich
    Commented Jan 21, 2015 at 9:58

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