helene_t, on Jul 19 2007, 01:54 AM, said:
P_Marlowe, on Jul 19 2007, 08:44 AM, said:
not wait for them.
An simple example: It is possible that you may hit the precise
center of a circular target with an arrow, but the probability
is zero.
No! Emphatically no! Suppose your digital ArrowHitMeter reports the distance from the center to your hit as 0.0000. Then all you know is that the distance was rounded of to zero by four digits after the period, i.e. it is between 0 and 0.00005. That range has positive width and presumably a positive probability. If the probability was zero it would not have happened.
This may sound like a silly measurement-technology problem but it's not. It's not even a physical problem, related to Heisenberg's uncertainty principle or some such. It's a fundamental principle in probability theory: If you have a random variable on a continous scale (such as distance from the center) the observations of that random variable are always ranges with positive width. You cannot pick a real number. You can pick an integer, or you can pick a range of real numbers.
Of course you can ask me to pick a "real" number and I'll be happy to think of sqrt(2) or pi or some such. But that's an illusion. I can only pick from the countable subset of the real numbers that can be expressed by mathematical formalism (or whatever language I think in). So effectively I'm picking an integer and then thinking of some transform of that integer which happens to be a non-integer number.
You are mistaken.
If probability of an event is zero, it does not mean it is impossible.