Probability of having K cigarettes in one of the 2 packs of cigarettes

Question

We have 2 packs of cigarettes, N cigarettes in each of the pack.

We keep taking out a cigarette from those packs randomly, until one of the packs gets empty.

How much is probability that the other pack contains exactly K cigarettes

Answer 1

This is almost but not quite Banach's matchbox problem. In that problem, the experiment continues until one of the packs is discovered to be empty upon an attempt to take a cigarette from it, whereas you want to end the experiment as soon as the pack gets empty.

If the other pack contains $K$ cigarettes, we had $N-1$ cigarettes taken from the empty pack and $N-K$ from the other pack, and then one cigarette from the empty pack. This can be done in $\binom{2N-K-1}{N-1}$ different ways that each have probability $\left(\frac12\right)^{2N-K}$, and either of the packs could have been the empty pack, which yields a factor of $2$. Thus the probability for the other pack to contain $K$ cigarettes is

$$\binom{2N-K-1}{N-1}\left(\frac12\right)^{2N-K-1}$$

for $0\lt K\le N$.