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Erik Iverson

2009-Aug-31 19:16 UTC

[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

Greg Snow

2009-Aug-31 19:55 UTC

head link

[R] Offtopic, HT vs. HH in coin flips

Well,

If the first flip is H, then the HT pattern occurs with the first flip in the
second run (after however long the 1st run of heads is).  If the first flip is
T, then the second run will be H's and the HT pattern will be the first flip
of the 3rd run.  So the HT pattern will occur after 1 or 2 runs (at the
beginning of the 2nd or 3rd).

On the other hand, the HH pattern can occur after any number of runs (given that
the H runs are only 1 long).

This means that the HH pattern has a higher probability of being in the right
tail of the distribution which will increase the mean.  The probability of HH or
HT as the 1st pair is the same.

Just looking at the first 3 flips, the probability of HH occurring first at
flips 2 and 3 has only 1 chance (THH, HHH means that the first HH was at 1 and
2) and therefore has probability 1/8.

The probability of HT first occurring at 2 & 3 has 2 options THT or HHT and
therefore is twice as likely.

Does this help?



-- 
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
801.408.8111

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
> project.org] On Behalf Of Erik Iverson
> Sent: Monday, August 31, 2009 1:17 PM
> To: r-help at 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 at 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.

David Winsemius

2009-Aug-31 20:11 UTC

head link

[R] Offtopic, HT vs. HH in coin flips

Case starting with H: Pr= 0.5
H first H second
Subcase  1a: Pr= 0.5 * 0.5 = 0.25
----
H first T second... leads to TH evenually
Subcase 1b: Pr = 0.5 * 0.5 = 0.25

==================Case T first: Pr = 0.5
all subcases lead to TH first


--  
David.

On Aug 31, 2009, at 3:16 PM, Erik Iverson wrote:
> 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 at 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.
David Winsemius, MD
Heritage Laboratories
West Hartford, CT

Erik Iverson

2009-Aug-31 20:35 UTC

head link

[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 at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Erik Iverson
Sent: Monday, August 31, 2009 2:17 PM
To: r-help at 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 at 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.

(Ted Harding)

2009-Sep-01 09:39 UTC

head link

[R] Offtopic, HT vs. HH in coin flips

On 31-Aug-09 19:16:33, Erik Iverson wrote:> 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
Be very careful about the statement of the problem!

[1] The probability that "HH" will occur first (i.e. before
"HT")
is the same as the probability that "HT" will occur first (i.e.
before "HH").

[2] However, the probability that the first occurrence of "HT"
will be on a given position of the "H" is generally not the same
as the probability that the first occurrence of "HH" will be on
the same position of the first "H".

[1]: At the first occurrence of (either "HH" or "HT"), there
is
an initial string S, ending in an "H", followed by either an
"H"
(for "HH") or a "T" (for "HT"). Both are equally
likely.

So the probability that the first occurrence of (either "HH" or
"HT")
is an ""HH" is the same as the probability that it is an
"HT".

[2]: (A) the first occurrence of an "HH" is in a sequence of
any collection of "H" and "T" provided there is no
"HH" in the
sequence, and the last is "H", followed by "H".
However, "HT" is allowed to occur in the sequence.

But (B) the first occurrence of an "HT" is in a sequence of
(zero or more "T") followed by (1 or more "H") followed by
"T".
This is the only pattern in which "HT" does not occur prior to
the final "HT".
Similarly, "HH" is allowed to pccur in the sequence.

The reason that, in general, the probability of "HH" first occuring
at a given position is different from the probability if "HT" first
occurring at that position lies in the differences between the
number of possible sequences satisfying (A), and the number of
possible sequences satisfying (B).

The first few cases ("HH" or "HT" first occurring at (k+1),
so
that the position of the first "H" in "HH" or "HT"
is at k) are,
with their probabilities:

k=1:       HH     HT
          1/4    1/4

K=2:      THH     HHT
                  THT
          1/8     2/8

k=3:     TTHH     HHHT
         HTHH     THHT
                  TTHT
         2/16     3/16

k=4:    TTTHH     HHHHT
        THTHH     THHHT
        HTTHH     TTHHT
                  TTTHT
         3/32     4/32

The "HT" case is simple:
  P.HT[k] = Prob(1st "HT" at (k+1)) = k/(2^(k+1))
Exercise for the reader: Sum(P.HT) = 1

The "HH" case is more interesting. Experimental scribblings on
parer threw up an hypothesis, which I decided to explore in R.
Thanks to Gerrit Eichner for suggestion the use of expand.grid()!

  ## Function to count sequences giving 1st HH on throw k+1
  countHH <- function(k){
    M <- as.matrix(expand.grid(rep(list(0:1),k)))
    ix <- (M[,k]==1) ## k must be an H (then k+1 will be H)
    for(i in (1:(k-1))){ ix<-ix&( !((M[,i]==1)&(M[,i+1]==1)) ) }
    sum(ix)
    ## list(Count=sum(ix),Which=M[ix,])
  }

Now, ignoring the case k=1:

  HHcounts <- NULL
  for(i in (2:12)){ HHcounts<-c(HHcounts,countHH(i)) }
  rbind((3:13),HHcounts)

  #         [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
  #            3    4    5    6    7    8    9   10   11    12    13
  #HHcounts    1    2    3    5    8   13   21   34   55    89   144

Lo and Behold, we have a Fibonnaci sequence! Another exercise for
the reader ...

Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 01-Sep-09                                       Time: 10:38:58
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