scott wrote:
> The mistake in your logic is assuming the probability of choosing 1 item
> from an infinite set is zero, it isn't, it is 1/infinity or "infinitely
> small".  In many cases this can be treated as zero, but when you start

	Nope. Probability theory gives zero. Not "arbitrarily small". The very
same mathematics that gives you a 1 for the monkey scenario gives you a
0 here. Which is why I keep saying they are identical.

	Rather, the "equivalent" argument to what you're saying is that the
probability of getting at least one tails in an infinite number of flips
is 1 - 1/infinity, and then me claiming that it isn't 1.

	As far as the math goes, there's no such thing as 1/infinity, nor is
there 1 - 1/infinity. The process to calculate the first gives you a 0,
the process to calculate the latter gives you 1.

> summing over an infinite number of items (ie what is the probability
> that I chose any of these items, or if I try an infinite times will I
> get this one?) there is an important difference.
> 
> The probability of you choosing 1.847 when asked to choose a number
> between 0 and 1 is really zero.  Even if you try an infinite number of
> times, it's still zero probability.
> 
>> Getting a sequence of all heads forever is identical to picking a point
>> from 0 to 1. Both have probability 0.
> 
> Mathematicians seem to disagree with you on that one.  1/infinity is not
> defined as zero (because it can often lead to problems like the above),
> however the infinite sum of 1/2+1/4+1/8+... (ie probability of getting
> no tails after infinite throws) is defined as 1.

	I don't see the logic. The only reason it is 1 is that after n flips,
the probability of getting a tails is 1 - (1/2)^n (which is the same as
your sum, which I could loosely (and meaninglessly) write as 1 -
1/infinity). Then they just take the limit to infinity to get 1. The
probability of getting all heads after n flips is (1/2)^n, which goes to
0 as you take the limit.

	Likewise, there is a 1-1 correspondence between each number from 0 to 1
and an infinite sequence of heads and tails. In other words, if you
state some number between 0 and 1, I can give you a unique sequence that
corresponds to that number (and vice versa). Hence, the probability of
picking that number is the same as the probability of getting that exact
sequence.

	Thus, if you state that the probability of getting all heads is 0, then
so is the probability of picking 0.5. Likewise, if you state that you
must get at least one tails, then I can likewise state that you must
pick some number _other_ than 0.5.

	When I say a 1-1 correspondence exists, I'm not saying it loosely, or
intuitively. I can rigorously show it to you: Just take the binary
expansion of any number from 0 to 1, and replace all 0's with heads and
all 1's with tails.

> Of course you can discuss how this relates to reality, but I'd rather
> not get involved in that one, it could an infinitely long time :-)

	But that *is* what Warp and I are complaining about. I don't disagree
with the math, and neither does he.


-- 
If a pig lost it's voice, would it become disgruntled?


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