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Old 2010-04-29, 16:23   #1
henryzz
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Default Coin tossing 2

Here is another look at coin tossing.
Which of these three strings is most likely to have been produced by 32 tosses of a coin and not constructed manually? Can anyone come up with some statistics to verify that?
a) THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH
b) THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT
c) TTHTTHHTHHTTTTTHTTHHTTHTTTTTTHTH
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Old 2010-04-29, 16:53   #2
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Quote:
Originally Posted by henryzz View Post
Here is another look at coin tossing.
Which of these three strings is most likely to have been produced by 32 tosses of a coin and not constructed manually? Can anyone come up with some statistics to verify that?
a) THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH
b) THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT
c) TTHTTHHTHHTTTTTHTTHHTTHTTTTTTHTH
They are all equally probable.

Last fiddled with by rogue on 2010-04-29 at 16:53
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Old 2010-04-29, 16:55   #3
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Each one has a 1/232 chance of being produced by 32 tosses of a coin. Thus none of them is more likely than the others to appear. C is the one that most *resembles* the majority of 32-toss sequences.
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Old 2010-04-29, 18:55   #4
henryzz
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I was hoping someone wouldn't come up with that answer.
Try this:
How probable is it that in a sequence of 32 tosses you would get runs of heads or tails of length r?

Then which do you think was not created manually.

Last fiddled with by henryzz on 2010-04-29 at 18:55
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Old 2010-04-29, 22:06   #5
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Quote:
Originally Posted by henryzz View Post
How probable is it that in a sequence of 32 tosses you would get runs of heads or tails of length r?
Do you mean at least a run of r or exactly r?

The exactly r would be much more difficult since that means the one before the first toss of the run must be the "other" side and the one immediately after the run must also be the "other" side.

But there is a slight bonus if the run starts at toss 1 or ends at toss 32.

Any takers?
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Old 2010-04-30, 16:59   #6
henryzz
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Quote:
Originally Posted by RichD View Post
Do you mean at least a run of r or exactly r?

The exactly r would be much more difficult since that means the one before the first toss of the run must be the "other" side and the one immediately after the run must also be the "other" side.

But there is a slight bonus if the run starts at toss 1 or ends at toss 32.

Any takers?
Don't mind which. If we solve the easy one though it would be worth trying the harder.
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Old 2010-05-13, 01:46   #7
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I think what henryzz wants to hear, is that since we expect N/2 runs of length 1, N/4 runs of length 2,...,N/2^k runs of length k, we could use chi-square to generate confidence intervals. A really cheap test could actually be to simply count runs R, because then some expression like (N-R-1)/(N-2) should be t-distributed.
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Old 2010-05-13, 02:55   #8
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Quote:
Originally Posted by __HRB__ View Post
we expect N/2 runs of length 1, N/4 runs of length 2,...,N/2^k runs of length k
May have too much blood in my alcohol stream ATM, but

1/2 + 2/4 + .. k/2^k +.... = 2

?

Sundance
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Old 2010-05-13, 04:05   #9
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Quote:
Originally Posted by __HRB__ View Post
A really cheap test could actually be to simply count runs R, because then some expression like (N-R-1)/(N-2) should be t-distributed.
OK, apparently the expressions are a little more complicated. According to:

https://pantherfile.uwm.edu/ericskey...L29/node2.html

If we let R denote the number of runs, H the number of Heads and T the number of Tails, then

E[R]=2*H*T/(H+T)+1

and

VAR[R]=2*H*T*(2*H*T-H-T)/[(H+T)^2*(H+T-1)]

and the standardized number of runs is asymptotically normal distributed. In any case, smaller standardized values should be considered 'preferred by henryzz'. I'm too lazy to plug in the actual numbers to give a definite answer.

Quote:
Originally Posted by davieddy View Post
May have too much blood in my alcohol stream ATM, but

1/2 + 2/4 + .. k/2^k +.... = 2

?

Sundance
Cheers!

Butch

Last fiddled with by __HRB__ on 2010-05-13 at 04:16
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Old 2010-05-14, 00:47   #10
Orgasmic Troll
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Quote:
Originally Posted by henryzz View Post
Here is another look at coin tossing.
Which of these three strings is most likely to have been produced by 32 tosses of a coin and not constructed manually? Can anyone come up with some statistics to verify that?
a) THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH
b) THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT
c) TTHTTHHTHHTTTTTHTTHHTTHTTTTTTHTH
just at a glance, C
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Old 2010-05-14, 11:21   #11
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With the greatest (lack of) respect, have any of you
ever considered why nCr = n!/((n-r)!*r!) ?
Or even glanced at the "coin tossing" thread?

David

Or heard of the binomial theorem,
let alone the binomial distribution?

Last fiddled with by davieddy on 2010-05-14 at 11:41
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