A smoker has two matchboxes: one in his right pocket and another in his left pocket, each of which contains $N$ matches. Each time he wants to smoke, he chooses (randomly) a pocket and lights a cigarette with a match taken from the corresponding matchbox and this process keeps going until he empties a matchbox. Determine the following probabilities
(a) By the time he empties the matchbox from the left pocket, the right pocket's matchbox contains $k$ matches, where $k\in\{1,2,\ldots,N\}$.
(b) By the time he empties the first matchbox, the other one has $k\in\{1,2,\ldots,N\}$ matches.
MY ATTEMPT
Let us define $X$ as the random variable which counts how many matches have been chosen from the left pocket and $Y$ as the random variable which counts how many matches have been chosen from the right pocket. They can be written as sum of independent Bernoulli random variables. More precisely, we have
\begin{align*} X = \sum_{k=1}^{N}X_{k}\quad\text{and}\quad Y = \sum_{k=1}^{N}Y_{k} \end{align*}
Thus I think the distributinos should be $X\sim\text{Binomial}(N,0.5)$ and $Y\sim\text{Binomial}(N,0.5)$, but I am not sure about this. Neither I know how to describe correctly the target event. Anyone could help me out? Thanks in advance!