Re: Coin tossing guessing strategy...

On Tue, 22 Apr 2008 03:55:24 -0400, quasi <quasi@xxxxxxxx> wrote:

On Tue, 22 Apr 2008 00:39:43 -0700 (PDT), Chula Pittayapinun
<pastelsalad@xxxxxxxxx> wrote:

On Apr 21, 6:19 am, quasi <qu...@xxxxxxxx> wrote:

On Mon, 21 Apr 2008 10:57:54 -0700 (PDT), Chula Pittayapinun wrote:

On Apr 21, 5:27 am, Ray Vickson <RGVick...@xxxxxxx> wrote:

ANY *particular* string, such as HTTHHHTHTHTT has the same probability
of occurrence as any other string of the same length, such as
HHHHHHHHHHHH or TTTTTTTTTTTT (strings of length 12 in this case). It
is true that strings of length 12 that have 5 heads and 7 tails are
much more probable than all H or all T, but that is not what we are
talking about here. You need a precise string in exactly one order,
because according to your description of the game, you lose as soon as
the element in the nth position (H or T) fails to match the actual
result of the nth toss, if you have not already lost before the nth
toss. Just guessing the right number of H's and T's is not good
enough.

That is what I also thought. But could you please elaborate on 'that
is not what we are talking about here'? I need a precise formal
argument, if possible, to explain to my friends. (One of them has
invoked the argument of random walk- that HHH...H has lower prob of
occurance than HTTHHHTHTT.)

Coins have no memory.

Moreover, a coin is blind.

How would a coin even know whether it came up H or T?

The simplest way to defeat these fools who think a fair coin is biased
with respect to sequences of n flips, is to note that if some sequence
of say 10 flips was more likely than some other sequence of 10 flips,
then a similar bias (possibly a little less) should hold for 9 flips,
right? In other words, it's not reasonable to claim the bias happens
only for n > 9. Once you convince your challenger that n = 9 would
also show a bias, then ask what about n = 8? When you reach n = 2, get
out some coins and experiment. Does your challenger really believe any
of the 4 sequences HH, HT, TH, TT is more likely than any other?

There's something dependent on the order of n in the argument of one-
dimensional random walk (please refer to wiki page of the topic
'random walk'), although I'm not sure how that argument is related to
this problem. Furthermore: 'for any random walk in one dimension,
every point in the domain will almost surely be crossed an infinite
number of times.' Does this mean that the longer the string (as n
approach /infty), it is more likely that the number of H's will equal
T's? If not, then why is it not applicable in this problem setting?

Read my prior explanation.

For 2 coins, do you think there a bias, even a slight bias with
respect to the 4 possible sequences HH, HT, TH, TT?

If not, then why would you think a bias suddenly develops for a longer
sequence, say a 10 coin sequence?

In fact, assuming a fair coin, all 10 coin sequences are equally
likely.

Don't confuse the equiprobability of n coin sequences with the
probability distribution for the _difference_ between the number of
heads and tails. Thus, for an n-term sequence, it's much more likely
that the difference is 0 than n,

Correction:

Of course, if n is odd, the difference can't be exactly zero.

In general the possible differences range from -n to n by 2.

A corrected statement is this ...

For an n-term sequence:

If n is even, then it's much more likely that
the difference is 0 than n.

If n is odd, and n > 1, then it's much more likely that
the difference is 1 than n.

game in question. You don't care about the totals. Effectively, you
are asking if one particular n coin sequence is more likely than
another. The answer is "no".

quasi
.

Relevant Pages

  • Re: Coin tossing guessing strategy...
    ... You need a precise string in exactly one order, ... with respect to sequences of n flips, is to note that if some sequence ... In fact, assuming a fair coin, all 10 coin sequences are equally ... heads and tails. ...
    (sci.math)
  • Re: Coin tossing guessing strategy...
    ... You need a precise string in exactly one order, ... with respect to sequences of n flips, is to note that if some sequence ... In fact, assuming a fair coin, all 10 coin sequences are equally ... heads and tails. ...
    (sci.math)
  • Re: Coin tossing guessing strategy...
    ... ANY *particular* string, such as HTTHHHTHTHTT has the same probability ... How would a coin even know whether it came up H or T? ... with respect to sequences of n flips, is to note that if some sequence ... it's not reasonable to claim the bias happens ...
    (sci.math)
  • Kolmogorov Complexity - again
    ... in which I proved that the probability for a given finite string not being ... the concepts of KC do indeed play a key role in the detection of bias ... with better than even odds of success - without knowing the actual ... characters in the sequence will share the same pattern increase. ...
    (talk.origins)
  • Re: Attention Sean - question about CSI
    ... that the detection of high CSI indicates is likely non-random bias. ... In short, if a string is algorithmically random, it is not predictable ... the reference sequence or one that is very far away. ...
    (talk.origins)
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