Can you clarify the part highlighted in red? Do you mean specifically to generate N random numbers from the uniform distribution between, say, 0 and L?
If I'm writing code for this experiment, I can't generate a random number without telling the computer precisely which probability distribution to use (and the corresponding distribution parameters). Most computer programs, for instance, will allow one to generate a random number uniformly between [0, L], but then you will need to supply the value for the upper limit L. (Reverend Bayes, is that you?)
Randomness comes in many forms (e.g., normal, uniform, exponential, etc. ) and I don't think it is possible to generate random numbers without specifying which probability distribution we are working with.
I'd love to hear about this experiment that does not depend on the probability distribution of how much money is in A and B. My feeling is that Reverends Bayes is hiding somewhere, possibly heavily disguised, in the set-up. But I may be wrong, I often am.