Re: [MLUG - DISCUSSION] fun math thing

[Mike Miller <EMAIL:PROTECTED>]

On Thu, 1 Jun 2006, Stephen Montgomery-Smith wrote:

> Mike Miller wrote:

> > I can see that.  A similar idea is that the probability is zero that 
> > the number will be rational.  I tell people this and they don't believe 
> > me. I will tell them that it is impossible for the randomly selected 
> > number to be rational and they will counter that the rational numbers 
> > are there, so it must be possible to pick one.  I guess it is 
> > counterintuitive for some people but it seems obvious to me.
>
> Well I would counter that they are correct - it is possible that the 
> random number is rational, just that the probability of it being so is 
> zero.
>
> I similar event which is theoretically possible but has probability zero 
> is to keep tossing a coin, and it never comes up with a head.


Never in a finite series of tosses.  I might not understand the definition 
of "theoretically possible" but it seems to me that it is theoretically 
impossible for a coin to always come up with a head.  I assume we are 
talking about a Bernoulli process with probability 1/2 of a head.

It is also theoretically impossible to pick at random a real number 
uniformly distributed on [0,1] and have it be a number whose digital 
(binary) representation can fit on a hard disk.  In fact, it wouldn't fit 
on all the hard disks on the planet put together.

Suppose we are to pick at random a real number uniformly distributed on 
[0,1].  Is it theoretically possible that we will chose 1/2?  I say that 
it is impossible.  It is also impossible that we will choose any number 
that we name in advance.  Furthermore, it is impossible that we will 
choose any one of the numbers on a list that we write in advance.  That 
list can even be infinitely long -- but it is a list, and therefore it has 
a countable infinity of elements (same cardinality as positive integers).

I guess I don't see how we can say that some event occurs with probability 
zero, and yet it is possible that it will occur.

Mike

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