Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he reaches into his pocket and discovers for the first time that the box picked is empty. If it is assumed that each of the matchboxes originally contained $N$ matches, what is the probability that there are exactly $k$ matches in the other box?
I guess the problem is quite known, see https://en.wikipedia.org/wiki/Banach%27s_matchbox_problem.
In the solution on wikipedia it says: "Without loss of generality consider the case where the matchbox in his right pocket has an unlimited number of matches".
Why can we simply assume this?
My naive guessing is something like that:
We can embed the sample space of the "matchbox experiment" into a bigger sample space, namely the one of a negative binomially distributed random variable, without changing the probabilities. E.g. Let $(L,L,R,R,L)$ denote one possible outcome where each matchbox contains $3$ matches. If I choose between both pockets one more time then $$P((L,L,R,R,L))=P((L,L,R,R,L))\cdot (P(L)+P(R))=P((L,L,R,R,L))\cdot 1=P((L,L,R,R,L)).$$ So the probibility of my original outcome hasn't changed. Keeping this in mind my idea would be to extend this argument to $\infty$ which means I keep on choosing between the two pockets.