Re: Probability in an infinite sample space

Robert Low wrote:

I'd settly for a rough idea of what you think 'infinitesimal' means,
and a way of calculating with these infinitesimals so that
if e is the probability of picking any given integer, the
limit as n tends to infinity of ne is 1.

OK. I'll give it another try.

Remember the formula for the probability in {1,2,3 ... , n} that an
arbitrary natural is divisible by a fixed natural a ? It's floor(n/a)/a
Which is equal to 1/a + eps(a,n) where eps(a,n) < a/n .

Now we say that for large n the eps(a,n) can be _neglected_. The point
is that we, as engineers and physicists, we with our limited measuring
and calculating devices, are incapable of "seeing" this eps(a,n) for n
beyond some value that we find "large". n is not a precisely specified
value, but we are quite certain that this will happen, sooner or later.
Then eps(a,n) = a/n cannot be distinguished from 0 , especially because
it is "mixed up" numerically with the much larger value 1/a . Thus the
"rough idea" of our infinitesimals is that they can be thrown away when
compared with "finite" real values. You asked for a rough idea ...

Han de Bruijn

.

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