--- In

primenumbers@yahoogroups.com, "John" wrote:

>

>

>

> --- In primenumbers@yahoogroups.com, "djbroadhurst" d.broadhurst@

wrote:

> >

> >

> >

> > --- In primenumbers@yahoogroups.com,

> > Chris Caldwell wrote:

> >

> > > if you roll four standard 6-sided dice and add up the resulting

values,

> > > what is the probability of rolling prime number?

> >

> > Help! I did this a dumb way, but got a simple answer:

> > (4+20+104+140+104+56+4)/6^4 = 1/3

> > Is there a smarter way, please?

> >

> > David

> >

>

> Not all that dumb in my book. In fact, it is the most logical. Just

work out the number of permutations for n=5,7,11,13,17,19,23 and add

them up, and divide by 1296.

>

It seems to me there is a very short proof available. The prime numbers

in the range (4,24) are precisely those equivalent to 1 or 5, modulo 6.

Now it can very easily be shown by mathematical induction (**) that the

6^N values (modulo 6) obtained by adding N numbers selected "one from

each" of N copies of the 6-moduli {0,1,2,3,4,5} are represented the same

number of times {i.e. 6^(N-1) times}.

It then follows trivially that the required probability is 2/6 = 1/3.

I have only run this "through my head", but it seems correct to me.

(**) I suspect it is a trivial consequence of some algebraic or

number-theoretic theorem.

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