I am studying the expected values of some specific slot machines where the number of consecutive chains of certain lengths affect the value, and although I am able to simulate their behaviour to compute a numerical expected value, I would be interested in showing mathematically where does that value come from. Essentially, I need to be able to answer the question:
In a binomial experiment with $n$ trials and a probability of success $p$, what is the probability to have $q$ chains of $r$ consecutive successes?
I have searched online and on math.stackexchange.com, but I have been unable to find a general solution to this problem.
Is there one?
Edit: Some clarifications
- Not all successes need to be part of a chain.
- A chain of $r$ successes must have exactly $r$ successes.
- If $r=3$, $SSSSSS$ would be considered to be two chains, but $SSSSSF$ would be considered to be only one.
Let's look at a specific numerical example.
If there are n=2000 trials and you want exactly q=5 chains, each chain being a word composed of exactly r=13 consecutive successes, then each word must be terminated by a failure. Thus we have the configuration
-W-W-W-W-W- where each word (chain) looks like W=SSSSSSSSSSSSSF and each - is a possible location to insert failures. The total number of characters in this sentence is 2000.
Each individual word W has probability $p^{13} (1-p)$.
The 5 words use up 5*14 =70 characters, so there are r=2000-70=1930 residual failures that need to be unloaded in any or all of the six dumping locations designated by the -. Strings of total length 2000 of this special type each have probability $[p^{13} (1-p)]^5 (1-p)^{1930}$.
To count the total number of such special strings of length 2000 that can be constructed, we must determine all ways that 1930 can be split as a sum of at most six non-negative integers (these integers specify the multiplicity within each of these six possible slots); ie. the number of compositions of 1930 into 6 nonnegative parts. This latter weak composition problem has a standard solution you can look up here: enter link description here
Specifically $C(1930+5,1930)$ Thus we recover the same result as that posted almost simultaneously by others.