With regards to your question, you have to make an assumption with regard to the probability.
Typically we do this in a couple of ways depending on the problem.
One way is to make mathematical assumptions and then derive the PDF (probability function) of the distribution. This is done with things like Binomial, Poisson, and other similar distributions.
The other way is to look at a distribution based on its fit to some model. We do this by either forcing a distribution to have a certain structure (like Normal, Chi-square, Uniform) or we can use what is called an empirical distribution which is just a fancy way of using the actual data from an actual experiment/process/etc and plotting a nice frequency histogram and normalizing it.
If you want to assume pure randomness then use a uniform distribution since it has the highest entropy, it gives the best model of randomness provided that all realizations are independent from the other ones.