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What's the probability that a random real(or natural) number is > 0. Me and a friend has been thinking about this question. It's an interesting question as there are lots of infinities involved. We intuitively think that it should be 50%, but we can't grasp the problem formally.

MelanzanaRipiena
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    The issue with this question is that it will only make sense assuming a specific probability distribution on the reals. For instance, you can assume that the "random real" is drawn from a Gaussian distribution with mean $0$, in which case it has indeed probability 1/2 to be positive. – Clement C. Aug 17 '15 at 17:09

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It depends on what you mean by "random". If you mean "uniformly distributed over $\Bbb{R}$" then there IS no uniform distribution over $\Bbb{R}$. If there was a uniform distribution, then $P([0,1])=P([1,2])=\dots\neq 0$. Same with $\Bbb{N}$. $P(\{1\})=P(\{2\})=\dots\neq0$. So we would get $P(\Bbb{R})=\infty$ or $P(\Bbb{N})=\infty$ which don't make sense.

You can choose other distributions, such as the normal distribution with mean $0$, then the probability is $1/2$. Or exponential distribution with probability $1$. Or delta distribution, or gamma or whatever. Distribution is important.