[r-t] random pn

[King, Peter R]

I presume here you are not bothered about truth, otherwise the mean must
be less than n! (any touch greater than n! must be false and so excluded
from the count, given that there must be true courses less than n! the
mean must come to less than n!). I think in terms of random walks it is
related to first passage problems (ie once you go beyond n! you must
stop searching.

> -----Original Message-----
> From: ringing-theory-bounces at bellringers.net 
> [mailto:ringing-theory-bounces at bellringers.net] On Behalf Of 
> Philip Saddleton
> Sent: 06 December 2004 19:41
> To: ringing-theory at bellringers.net
> Subject: Re: [r-t] random pn
> 
> 
> Richard Smith wrote:
> 
> >Chris Poole wrote:
> >
> >  
> >
> >>Here's a Friday teaser - suppose we're looking at methods 
> on N bells.  If
> >>I just write down random pieces of valid place notation, 
> and look at the
> >>method that is the outcome of these ordered bits of pn, 
> when do I expect
> >>it to come round?
> >>    
> >>
> >
> >Empirically, the answer seems to be about N!.  Here are the
> >results of 1000 random walks on 3 <= N <= 8 bells:
> >
> >  Bells  Mean         S.D.
> >    3    6.024   +/-  6.11812
> >    4    23.451  +/-  32.8958
> >    5    117.524 +/-  160.984
> >    6    729.176 +/-  927.313
> >    7    5367.71 +/-  6063.72
> >    8    40103.5 +/-  4.2949e+06
> >
> >  
> >
> The mean is precisely n!
> Take any sequence containing m different elements, and join 
> up the ends. 
> The mean distance between occurrences of the same element is 
> m (this is 
> independent of the length of the sequence or the distribution of the 
> individual elements).
> 
> I initially thought Chris was asking a completely different question: 
> pick a random lead head, and what is the mean number of leads 
> in a course?
> 
> -- 
> pabs
> 
> 
> 
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