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 $RRR$.
- 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?
An MO user had pointed out that a method known as the transfer-matrix method could be used to solve the problem. Quoted (almost) verbatim:- "The idea is to keep track of the colors of the last two balls drawn. One 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 one wants can be obtained by extracting coefficients from powers of a 4 by 4 matrix, and from this matrix one can get a rational generating function for the numbers." I did read up on this method in R. Stanley's Enumerative Combinatorics. However, I couldn't understand how it works, much less how it could be applied to this problem. Can anyone help me out with this?
- Even if the probability distribution is not easily attainable, an MO user pointed out that calculating the expectation value, variance and other moments is rather standard and that one could obtain an asymptotic central limit result for this problem. Could anyone shed light on how this could be done?
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.