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 2010-04-18, 02:05 #1 davieddy     "Lucan" Dec 2006 England 145128 Posts 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
2010-04-18, 02:13   #2
lfm

Jul 2006
Calgary

52×17 Posts

Quote:
 Originally Posted by davieddy 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 ?

2010-04-18, 02:46   #3
davieddy

"Lucan"
Dec 2006
England

145128 Posts

Quote:
 Originally Posted by lfm 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

 2010-04-18, 02:52 #4 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts 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
 2010-04-18, 03:13 #5 CRGreathouse     Aug 2006 3·1,993 Posts Nitpick: I would expect 999994/128; am I wrong?
 2010-04-18, 03:19 #6 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 2×23×137 Posts 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.
2010-04-18, 03:47   #7
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by retina 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

 2010-04-18, 04:14 #8 davieddy     "Lucan" Dec 2006 England 2·3·13·83 Posts 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
 2010-04-18, 11:45 #9 davieddy     "Lucan" Dec 2006 England 194A16 Posts 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
 2010-04-18, 13:57 #10 wblipp     "William" May 2003 New Haven 23·103 Posts 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.
 2010-04-18, 23:26 #11 davieddy     "Lucan" Dec 2006 England 2·3·13·83 Posts We make a great "double act" 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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