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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:

  1. The fraction of tails that are in sections of size three or more.
  2. The expected size of sections with tails.

I'd be very thankful if you could help me with it!

1 Answers 1