Re: [MLUG - DISCUSSION] fun math thing

[Stephen Montgomery-Smith <EMAIL:PROTECTED>]

Mike Miller wrote:
> 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

It really depends upon what you mean by "theoretically possible."  It is 
best to stick to the phrase "probability zero" and not get into the more 
philisophical discussion of whether it is possible or not.

Another point of discussion - it is theoretically a certainty that if 
you give a monkey a typewriter, and an infinite life, that eventually it 
will type the complete works of Shakespeare.  But it is practically 
impossible that this will ever happen.

Stephen

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