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Old 2010-04-18, 02:05   #1
davieddy
 
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Default coin tossing

A run of 5 "heads" is defined as the sequence THHHHHT.

How many such runs would you expect in a million tosses?

David
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Old 2010-04-18, 02:13   #2
lfm
 
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Quote:
Originally Posted by davieddy View Post
A run of 5 "heads" is defined as the sequence THHHHHT.

How many such runs would you expect in a million tosses?

David
a quick guess :

999993/128 ?
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Old 2010-04-18, 02:46   #3
davieddy
 
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Quote:
Originally Posted by lfm View Post
a quick guess :

999993/128 ?
That's what I think.

So counting heads (half a million):

1/8 + 2/16 + 3/32 + 4/64 +.... should = 1/2

Does it?

Last fiddled with by davieddy on 2010-04-18 at 02:54
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Old 2010-04-18, 02:52   #4
davieddy
 
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Yes

And the number of runs of 5 or more heads is

million/64

Last fiddled with by davieddy on 2010-04-18 at 03:32
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Old 2010-04-18, 03:13   #5
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Nitpick: I would expect 999994/128; am I wrong?
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Old 2010-04-18, 03:19   #6
retina
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Did you forget about THHHHHTHHHHHT? And THHHHHTHHHHHTHHHHHT? etc.

edit: nvmd, CRGreathouse got it.

Last fiddled with by retina on 2010-04-18 at 03:22 Reason: CRGreathouse got it.
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Old 2010-04-18, 03:47   #7
davieddy
 
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Quote:
Originally Posted by retina View Post
Did you forget about THHHHHTHHHHHT? And THHHHHTHHHHHTHHHHHT? etc.

edit: nvmd, CRGreathouse got it.
We didn't "forget" it.

My line of thinking was that each T had a probability of 1/64 of
being followed by HHHHHT and I expected 500,000 tails.

Last fiddled with by davieddy on 2010-04-18 at 04:00
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Old 2010-04-18, 04:14   #8
davieddy
 
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Default Pedantry

I think the million tosses should be prefixed and
postfixed with half a tail.

David

Last fiddled with by davieddy on 2010-04-18 at 04:17
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Old 2010-04-18, 11:45   #9
davieddy
 
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Default GIMPS Lucky Streak

As you know, Wagstaffe expects a ratio of 1.48 between
exponents of successive Mersenne primes.
The probabiliy of the "gap" being greater than this is 1/e.
The "Half Gap" when the probability is 50-50 is smaller (say1.3).
The mean of gaps smaller than the halfgap is less than sqrt(1.3).

A neat way of describing the lucky streak M40-M47 is that we have hit
a run of 7 gaps smaller than the "Half Gap".

From our previous deliberations, we expect this to occur once in
512 Mersenne primes.
We expect runs of 7 or more gaps less/greater than the Halfgap
to occur once in 128 Mersenne primes.

Our find is not that freakish!

David

Last fiddled with by davieddy on 2010-04-18 at 12:21
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Old 2010-04-18, 13:57   #10
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I agree with CRGreathouse. davieddy's analysis method works with the additional observation that is doesn't apply to tails in the last 6 tosses. An alternative method is to treat it as an 8 state Markov process on the states
Not-Started
Tail
One Head
Two Heads
Three Heads
Four Heads
Five Heads
Finished

With Intial probability "not started = 1". On the seventh step this reaches steady state probability. For that and all following steps, the probability of being in the "Finished" state is 1/128. For the first six steps it is 0.
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Old 2010-04-18, 23:26   #11
davieddy
 
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Who was it who said that there was no problem
so complicated that with a little ingenuity
couldn't be made more complicated?

David
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