SPUG: Stick Riddle
Wegley, Harry L
harry.l.wegley at pss.boeing.com
Fri Jan 3 13:27:38 CST 2003
In the real world, wouldn't the breaks occur at the molecular level. There are not an infinite number of molecules between the ends of the stick, so the probability of breaking the stick in 2 equal parts appears to be small, but finite. However, there are the problems of the geometry as well as the physical properties of the stick (e.g. it probably would not break normally to its surface; the stick is not a perfectly circular rod; molecules splinter off when it breaks, etc.) In the real world things usually become finite, but other complicating factors are introduced.
--Harry Wegley
-----Original Message-----
From: Jonathan Gardner [mailto:jgardn at alumni.washington.edu]
Sent: Friday, January 03, 2003 6:19 AM
To: Creede Lambard; Jeremy Calvert
Cc: spug-list at pm.org
Subject: Re: SPUG: Stick Riddle
On Thursday 02 January 2003 10:41 pm, Creede Lambard wrote:
> On Thu, 2003-01-02 at 22:23, Jeremy Calvert wrote:
> > > > > ... if A + B = C ...
> > > >
> > > > ... the probability of this happening is 0 ...
> > >
> > > Not so. If the first break results in two sticks of
> > > equal length ...
> >
> > Is so so:). The probability of your counter-example
> > happening is also 0. Otherwise, what is the
> > probability that the first break results in two sticks
> > of equal length?
>
> I still respectfully state that it's not so. The probability of the
> first break being exactly in the middle of the stick is vanishingly
> small, but so is the probability that the break appears anywhere else in
> the stick. Does that mean the stick can never be broken, since the
> probability of it breaking at any particular point is the same as the
> probability of it breaking at the exact center? (This probably has
> something to do with dividing by zero.)
>
> Not to mention the many, many circumstances in which the first break
> appears at location $x along the stick, and the second break causes the
> longer of the two sticks to split such that the longest piece left over
> has length of .5 .
>
> This is obviously a problem for some sort of calculus, but I'm afraid my
> calculus years are, ahem, many years behind me.
>
The chances of the stick breaking exactly in some predefined way is
infinitessimally small -- hence zero. My reasoning is as follows. A
mathematical stick can be cut anywhere between the two ends. There is an
infinite number of positions in which it can be cut, and so the chances for
any one of those positions to be cut is infinitessimally small.
This is not so when using perl's scalars. There is a finite number of places
you can make a cut along the stick, even if you use floating point numbers
(which aren't mathematically real numbers but give us a pretty good
approximation). If you were able to randomly generate *any* real number
between 0 and 1 in perl (you can't, so don't try!), then the situation for
the mathematical stick applies.
A stick that is real -- err, I mean a real stick, not a mathematical or perl
one -- is made up of atomic matter that can only be subdivided in a finite
number of ways. So, like the perl sticks, it can be broken in some predefined
way. The chances are very, very small, but not zero.
--
Jonathan Gardner
jgardn at alumni.washington.edu
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