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Say we flip a biased coin such that the probability of getting the same outcome in a row (head-head or tail-tail) is $p$.

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...

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.
  • 1
    If the probability of getting heads is denoted $q$ we have $$p=q^2+(1-q)^2.$$ Solve as a quadratic in $q$, and then this problem is reduced to some more standard ones.2012-03-04
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    @anon: Can you please add that as an answer?2012-03-04
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    i m not sure i get it. Here when q=1/2, p=1/2. Once we get heads, the prob that the next one is heads is p. So the prob of getting heads=tail can be 1/2 but p doesnt have to be 1/2. or maybe I didnt fully understand your answer.2012-03-04
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    *"Once we get heads, the prob that the next one is heads is p"* No, it is $q$. The probability of getting two heads in a row, where the first head is **not** taken for granted, **plus** the probability of getting two tails in a row (again, neither taken for granted), is $p$. @Aryabhata: I don't consider my comment an answer, only how to start off.2012-03-04

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