# Relation between repeat number in coin toss

I am trying to establish correlation between tossing of coins and occurring of repeats.

Coin is flipped 10 time as follows:

$${\rm H.T.H.H.H.T.H.T.T.T. }$$

After each repeat occurring I have put (R) as follows:

$${\rm H.T.H.H(R).H(R).T.H.T.T(R).T(R). }$$

So in this exercise I have 4 repeats, any patteren/symmetry/correlation could be estblised here?

Is it fair to conclude in 10 toss I get 4 repeats and establish for 12th or 13th flip repeat is certain?

(Stats novice)

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I made an edit to enhance readability. Please make sure I did not change the question semantics. – user2468 Apr 7 '12 at 4:08
Certainty is not achievable. It is possible, though highly unlikely, that the first $50$ tosses will monotonously alternate between H and T. But information about the distribution of the number of repeats is obtainable, and information about the mean number of repeats in say $10$ tosses (or $n$ tosses) tosses is easy to get. The mean number of repeats in $10$ tosses is $4.5$. – André Nicolas Apr 7 '12 at 4:18
Thanks Andre, I have tried in Excel using random command for 2000 flips. The R was 969, Heads were 984 and Tails were 1016. Now is is possible to get clear picture?Thanks in advance Rodney – Rodeny Apr 7 '12 at 5:36

We assume the coin is fair. Then after the first toss, the probability of a repeat is $1/2$. For $k>1$, the event there is an R at position $k$ is independent of previous locations of the R's. So if we toss the coin $n$ times, the number of R's has binomial distribution, where the number of trials is $n-1$, and the probability of success each time is $1/2$. Thus if $X$ is the total number of R's, then $$P(X=m)=\binom{n}{m}\left(\frac{1}{2}\right)^{n-1}.$$