 Solution

Use the positive-support geometric distribution, $\mathbb{P}(X=j)=p(1-p{)}^{j-1}$ for $j\ge 1$, $0<p\le 1$. Differentiating the geometric series suppliesFor the jar, let $W_r$ be the waiting time to remove one red ball when exactly $r$ red balls remain. Every draw has success probability $r/n$, so $W_r$ is geometric with that parameter. The $W_r$ are independent: after each success, fresh independent draws begin a new stage, and its conditional waiting distribution depends only on the deterministic count $r$, not on earlier waiting times. Replacing a green ball leaves that stage's count unchanged.

Thus the coupon collector problem decomposition is $T={\sum }_{r=1}^n{W}_r$, andThis is the coupon collector waiting-time variance. For $n=1$, the formula correctly gives zero variance and a deterministic one-minute completion.