Consider the case of an infinite (or finite $n$) string of coin tosses, and let $q$ and $1-q$ be the probabilities that the coin comes up tails and heads, respectively. (For simplicity, we can take $q=\frac12$ so that the fraction of tails and heads are the same.)
Let $p$ be the probability that, if a given toss is tails, it will be followed by tails. (If $q = \frac12$, this is the same probability as for getting heads after heads, might as well be a different probability though.)
What is the probability of getting three or more tails consecutively out of $n$ flips (and alternatively out of infinite number of flips).
For example: $TTT-H-TTT-HH-TTTT\dots$
We can have half tails and half heads in total (when $q=\frac12$). $p$ does not affect the fraction of tails and heads in the limit, but affects how they are clustered. So when $p=1$ we have only 2 separate 'sections' such that one section is composed of only tails and the other runs of only heads. (ex: $TTTTT \dots HHHHH$). For $p=0$, it will be like $THTHTHTH\dots$
I am looking for:
- The fraction of tails that are in sections of size three or more.
- The expected size of sections with tails.
I'd be very thankful if you could help me with it!