For those interested, the original puzzle was detailed in a TED presentation
by Peter Donnelly, which you can see at blog.revolution-computing.com here:
http://bit.ly/2l0ZwS
The original problem was: which coin-toss sequence do you expect to see
first, HTH or HTT? Peter's explanation comes at about 5:00 in the video, and
for a lay audience it's great. (The champagne metaphor is wonderful.)

I'm loving the R simulations of this and related problems in this thread!

Cheers,
# David

On Mon, Aug 31, 2009 at 1:55 PM, William Dunlap <wdunlap@tibco.com> wrote:

> It gets even more interesting when you ask about which
> of 2 triples of head/tail sequences appears first in an
> infinite sequence of heads and tails.  Martin Gardiner
> wrote about this in the early 1970's
>  Martin Gardner, "Mathematical Games: The Paradox of the Nontransitive
> Dice and the Elusive Principle of Indifference." Scientific American
> 223, 110-114, Dec. 1970
> (and perhaps again in 1974).  His book, "The Colossal
> Book of Mathematics: classic puzzles, paradoxes, and
> problems" has that stuff reprinted and updated.
>
>
> Bill Dunlap
> TIBCO Software Inc - Spotfire Division
> wdunlap tibco.com
>
> > -----Original Message-----
> > From: r-help-bounces@r-project.org
> > [mailto:r-help-bounces@r-project.org] On Behalf Of Erik Iverson
> > Sent: Monday, August 31, 2009 1:35 PM
> > To: Erik Iverson; r-help@r-project.org
> > Subject: Re: [R] Offtopic, HT vs. HH in coin flips
> >
> > Part of my issue was that I was not answering my original
> > question.  "What is more likely to show up first, HT or HH?"
> > The answer to that turns out to be "neither", or "identical chances".
> >
> > ht <- replicate(2500,
> >                 paste(sample(c("H", "T"), 100, replace = TRUE),
> >                       collapse = ""))
> >
> > hts <- regexpr("HT", ht) + 1
> > hhs <- regexpr("HH", ht) + 1
> >
> > ## which is first?
> > table(hts < hhs)  # about 50/50
> >
> > summary(hts)      #mean of 4
> > summary(hhs)      #mean of 6
> >
> > So, "What is more likely to show up first, HH or HT?" is of
> > course a different question than "Are the expected values of
> > the positions for the first HT or HH the same?"  I suppose
> > that's where confusion set in.  It seems that if HH appears
> > later in the string on average (i.e., after 6 tosses instead
> > of 4), that the probability of it being first would be lower
> > than HT, which is obviously wrong!
> >
> > A quick graphic that helps show this (you must run the above
> > code first):
> >
> > library(lattice)
> >
> > ht.df <- data.frame(count = c(hts, hhs),
> >                     type = gl(2, 1250, labels = c("HT", "HH")))
> >
> > barchart(prop.table(xtabs(~ count + type, data = ht.df)),
> >          stack = FALSE, horizontal = FALSE,
> >          box.ratio = .8, auto.key = TRUE)
> >
> > Thanks to all those who replied, and also someone sent me the
> > following link off list, it also clears up the confusion:
> >
> > http://www.mit.edu/~emin/writings/coinGame.html
> >
> > Best,
> > Erik
> >
> > -----Original Message-----
> > From: r-help-bounces@r-project.org
> > [mailto:r-help-bounces@r-project.org] On Behalf Of Erik Iverson
> > Sent: Monday, August 31, 2009 2:17 PM
> > To: r-help@r-project.org
> > Subject: [R] Offtopic, HT vs. HH in coin flips
> >
> > Dear R-help,
> >
> > Could someone please try to explain this paradox to me? What
> > is more likely to show up first in a string of coin tosses,
> > "Heads then Tails", or "Heads then Heads"?
> >
> > ##generate 2500 strings of random coin flips
> > ht <- replicate(2500,
> >                 paste(sample(c("H", "T"), 100, replace = TRUE),
> >                       collapse = ""))
> >
> > ## find first occurrence of HT
> > mean(regexpr("HT", ht))+1    #mean of HT position, 4
> >
> > ## find first occurrence of HH
> > mean(regexpr("HH", ht))+1    #mean of HH position, 6
> >
> > FYI, this is not homework, I have not been in school in
> > years.  I saw a similar problem posed in a blog post on the
> > Revolutions R blog, and although I believe the answer, I'm
> > having a hard time figuring out why this should be?
> >
> > Thanks,
> > Erik Iverson
> >
> > ______________________________________________
> > 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.
> >
>
> ______________________________________________
> 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.
>



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