Probability function over infinite set

Hmm, I was just talking to a friend of mine...and I said that

Personally I would like to define the discrete probability function to be $|event|\over |sample space|$

Then I gave an example about rolling a fair die P(outcome is even)=$|2,4,6| \over |1,2,3,4,5,6|$ which is 50%, and my friend asked me if it worked with infinite discrete set...without thinking too much I said...of course, if the numerator is all even positive integers and the denominator is all positive integers, then I know the probability is going to be 50%

but later, when I tried to prove it mathematically...I failed, since I know that set of even positive integers and all positive integers have the same cardinality...the one to one mapping is just times two

so...my function should return me 1? I know this is not making sense...can someone help me...I think somehow I confused myself :(

• are you sure it is 50%? is the die fair? – BCLC Jan 31 '16 at 17:51
• @BCLC, good question I think this probability function will only work when each element in the sample space is equally likely, if this is not the case...maybe we need to introduce a gcd? – watashiSHUN Jan 31 '16 at 20:44

• What I am saying is that there is no way to define a pdf so that every integer has equal weight. We can define a density on the positive integers by something like $p(1) = 1/2$, $p(2) = 1/4$, $p(3) = 1/8$, and so on. – user296602 Jan 31 '16 at 23:50