There are 2 buckets. Each bucket has $n$ balls. You pick balls at random out of each bucket. You choose a bucket randomly to take a ball out of. What is the expected number of balls remaining in the non-empty bucket when the other bucket is empty? Assume $n \gg 1$
Let $Y$ denote the number of balls in the remaining bucket. It could have $1,2,\ldots, n$. I believe $Y$ follows a binomial distribution. But I am lost on how to represent the binomial distribution.
What is the number of independent trials here? At first, I thought it was the number of balls removed from both buckets, but for each $Y$ value, there is a different number of trials. For example, when $Y = 1$, this means we removed $2n - 1$ balls, so $2n-1$ is the number of trials. In general, for $Y = k$, $2n - k$ is the number of trials. The number of successes is $k$. But with this approach, the number of trials and successes is dependent on $Y$. Is there a better way to define the successes and number of trials for this problem?