Let's say that instead of a 2 sided coin, I have a three state
random generator, which generates each state with equal probability.

        Now I had already shown that the likelihood of getting a
_particular_ infinite sequence was the same as getting a _particular_
value between 0 and 1. (That is, the likelihood is zero - I'm arguing
purely mathematically - let's put the "real" world aside).

        Intuitively, that's because a point is infinitely smaller than
the interval [0,1].

        In fact, if I were to ask the probability of getting a rational
number, the answer is still 0. While the set of rationals is infinite,
it's still infinitely smaller than the unit interval - the latter is of
a higher infinity.

        So how about this one: In my three state random sequence
generator, what is the probability of getting *any* sequence that
doesn't have one of the states? If I were to label the states 0, 1 and
2, the question would be: What is the probability of getting a sequence
that doesn't have 1 in it?

        Now this is equivalent to asking: What is the probability that
if I pick any number at random from [0,1], the number will not have a 1
in it when written with a base 3 expansion?

        It may not be obvious what the solution is. The set of all such
sequences may appear small, but in one sense it isn't: It has the same
cardinality as the real line (it is of the same infinity as [0,1]).

        In point of fact, it's the Cantor set (i.e. what is the
probability that a number you pick will be from the Cantor set?):

http://en.wikipedia.org/wiki/Cantor_set

        Now I'll re-emphasize: While the Cantor set _appears_ to be
small, it is of the same cardinality as [0,1].

        It turns out this set has "measure" 0. And any integral of a
function (the uniform distribution in this case) over a set of measure
zero is 0.

        So the probability of getting any infinite sequence that doesn't
have one of the states is zero. I'm willing to bet that this generalizes
to any base.

        I think it's a cool result - shows that probability theory is at
least consistent.


-- 
Guitar for sale. Very cheap. No strings attached.


                    /\  /\               /\  /
                   /  \/  \ u e e n     /  \/  a w a z
                       >>>>>>mue### [at] nawazorg<<<<<<
                                   anl

Post a reply to this message