BonJoviJones
11-19-2002, 11:41 PM
So in the "basic question" thread below, the conversation veered to when P(X)=1 and P(X)=0 aren't certainties given infinite sets to work with. I'm intrigued and curious.
Here's my informal proof, anybody want to comment?
(I should note that this started as a "I don't get this post", but half-way through I think I figured it out. I'm quite proud /forums/images/icons/wink.gif)
A guy has a box of colored balls. There is one red ball, and the rest are yellow. If N is the number of balls in the bag, and R is picking a red ball:
P(R) = (lim n->inf) 1/N
This means that P(R)=0 _but_ there is still a red ball in there somewhere, and it can still be drawn.
Am I on track so far?
So let's expand to BruceZ's number example, which I'll rephrase as:
Given an infinite precision random number 0 to 1 inclusive, what is P(Rational Number)?
Using the above ball problem as a template, we get:
P(R) = (rational numbers)/(irrational numbers)
(My calc skills are breaking down, but we want the limit as rational numbers approach infinity, and as irrationtals approach infinity _faster_.)
Given that irrationals are uncountably infinite, while rationals are merely countably infinite it should boil down to P(R) = 0, but like the ball problem it's still possible to get a rational number.
How's that?
Here's my informal proof, anybody want to comment?
(I should note that this started as a "I don't get this post", but half-way through I think I figured it out. I'm quite proud /forums/images/icons/wink.gif)
A guy has a box of colored balls. There is one red ball, and the rest are yellow. If N is the number of balls in the bag, and R is picking a red ball:
P(R) = (lim n->inf) 1/N
This means that P(R)=0 _but_ there is still a red ball in there somewhere, and it can still be drawn.
Am I on track so far?
So let's expand to BruceZ's number example, which I'll rephrase as:
Given an infinite precision random number 0 to 1 inclusive, what is P(Rational Number)?
Using the above ball problem as a template, we get:
P(R) = (rational numbers)/(irrational numbers)
(My calc skills are breaking down, but we want the limit as rational numbers approach infinity, and as irrationtals approach infinity _faster_.)
Given that irrationals are uncountably infinite, while rationals are merely countably infinite it should boil down to P(R) = 0, but like the ball problem it's still possible to get a rational number.
How's that?