First of all, we must define what is a run of length r: is it a tail, then EXACTLY r heads and a tail again or is it AT LEAST r heads. Let's assume that we are looking for a run of EXACTLY r heads (and we toss the coin n times). Let X[1],X[2],...,X[n-r+1] be random variables such that Xi = 1 if there is a run of r heads starting at place i and 0 otherwise. Then the number of runs of length r is just sum(X), so the expected number of runs of length r is sum(E(X)) = sum(P(X[i]=1)). Now, for i=2,3,...,n-r P(X[i] = 1) = (1-h)*h^r*(1-h) and also P(X[1] = 1) = h^r*(1-h) and P(X[n-r+1] = 1) = (1-h)*h^r, so that the expected number of runs of length r is (1-h)*h^r*(2 + (n-r-1)*(1-h))
As to the distribution of the number of such runs, this is a much more difficult question (as mentioned by some other people). --- On Fri, 10/10/08, Harvey <[EMAIL PROTECTED]> wrote: > From: Harvey <[EMAIL PROTECTED]> > Subject: [R] runs of heads when flipping a coin > To: r-help@r-project.org > Received: Friday, 10 October, 2008, 4:16 AM > Can someone recommend a method to answer the following type > of question: > > Suppose I have a coin with a probability hhh of coming up > heads (and 1-hhh > of coming up tails) > I plan on flipping the coin nnn times (for example, nnn = > 500) > What is the expected probability or frequency of a run of > rrr heads* during > the nnn=500 coin flips? > Moreover, I would probably (excuse the pun) want the answer > for a range of > rrr values, for example rrr = 0:50 > > Of course I am more interested in an analytical solution > than a monte carlo > simulation solution. > > Thanks in advance, > > Harvey > > * OR "a run of rrr heads or more ..." > > [[alternative HTML version deleted]] > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, > reproducible code. ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
