Originally posted by Agerg
I tend to steer clear of stats and probability but since we're talkng about the real world and not an idealised mathematical one, don't we need to invoke conditional probabilty here?
Ie: what is the probability that a person wll pick a rational number in [0,1] given some finite upper bound for the amount of time a person can spend trying to express any number (before they die, say!)
In that case wouldn't you have to do the same with all probability problems involving humans? Mathematics is always about idealizations because if we are truly interested in the real world you wouldn't solve any problem you'd just end up categorizing all the variables that might affect the outcome of the
experience and these, in principle, are infinite.
Th problem with the example I gave is more or less what KN said. The thing is that most people you'd encounter on the street are exactly fluent in the language of mathematics and would choose rational numbers because they are more used in dealing with rational numbers than with rational numbers.
For this problem to be unbiased you'd have to be biased on your sample: you'd have to choose people that are as used with dealing with rational numbers as they are used in dealing with irrational numbers: mathematicians,physicists and engineers come to mind. And even still I suspect more than 50% would choose rational numbers.
Anyway my example was just to illustrate a very counter-intuitive result of real analysis (measure theory if you prefer) even though the set of rational numbers and the set of irrational numbers have an infinite amount of elements in a very precise sense the set of irrational numbers have infinitely more elements than the set of rational numbers.
And all of this is totally independent of the the trivial result that any event in a sample space that is continuous has a 0 probability of happening and yet these events happen all the time. (Assuming, of course, that the probability density function is well behaved).