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 :(

  • $\begingroup$ are you sure it is 50%? is the die fair? $\endgroup$ – BCLC Jan 31 '16 at 17:51
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    $\begingroup$ @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? $\endgroup$ – watashiSHUN Jan 31 '16 at 20:44

There is no uniform density function on a countable set such as the integers - that is, there is no way to define a probability distribution for which every integer is an equally likely output.

But a related concept is natural density or asymptotic density, which will (in some sense) capture the intuition that the evens and odds pair up as they do for finite sets.

  • $\begingroup$ I don't really get the first sentence, are you saying that we can't define a probability function over countable set like integers because of non-uniform density function(what does it mean...)? can you give me a counter example? ie something with a uniform density function and what would the probability function look like? thanks :) $\endgroup$ – watashiSHUN Jan 31 '16 at 21:00
  • $\begingroup$ 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. $\endgroup$ – user296602 Jan 31 '16 at 23:50

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