[R] runs of heads when flipping a coin
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.
Re: [R] runs of heads when flipping a coin
You can use 'sample' and 'rle': x - sample(c(H,T), 500, replace=TRUE, prob=c(.95, .05)) as.data.frame(unclass(rle(x))) lengths values 1 12 H 21 T 3 10 H 41 T 55 H 62 T 7 26 H 81 T 9 17 H 10 1 T 11 54 H 12 1 T 13 23 H 14 1 T 15 7 H 16 1 T 17 2 H On Thu, Oct 9, 2008 at 1:16 PM, Harvey [EMAIL PROTECTED] wrote: 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. -- Jim Holtman Cincinnati, OH +1 513 646 9390 What is the problem that you are trying to solve? __ 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.
Re: [R] runs of heads when flipping a coin
It seems to me that you are asking for: hhh - 0.1 sum(dbinom(x=0:50,size=500,prob=hhh)) [1] 0.5375688 You can use 'sample' and 'rle': x - sample(c(H,T), 500, replace=TRUE, prob=c(.95, .05)) as.data.frame(unclass(rle(x))) lengths values 1 12 H 21 T 3 10 H 41 T 55 H 62 T 7 26 H 81 T 9 17 H 10 1 T 11 54 H 12 1 T 13 23 H 14 1 T 15 7 H 16 1 T 17 2 H On Thu, Oct 9, 2008 at 1:16 PM, Harvey [EMAIL PROTECTED] wrote: 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. -- Jim Holtman Cincinnati, OH +1 513 646 9390 What is the problem that you are trying to solve? __ 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.
Re: [R] runs of heads when flipping a coin
On 10/10/2008, at 6:16 AM, Harvey wrote:
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.
I think this is a relatively deep problem analytically, and there
appears to
be a substantial amount of literature. Googling on ``probability of
a sequence
of runs'' got a lot of hits, including a pointer to
http://mathworld.wolfram.com/Run.html
which gives some interesting discussion and provides a number of
references.
(Feller volume 1 seems to be a good place to start, as usual.)
cheers,
Rolf Turner
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Re: [R] runs of heads when flipping a coin
IIRC this is a standard problem in digital communications theory, so you might want to look for literature in that arena. Carl 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. I think this is a relatively deep problem analytically, and there appears to be a substantial amount of literature. Googling on ``probability of a sequence of runs'' got a lot of hits, including a pointer to http://mathworld.wolfram.com/Run.html which gives some interesting discussion and provides a number of references. (Feller volume 1 seems to be a good place to start, as usual.) __ 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.
Re: [R] runs of heads when flipping a coin
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.
