Inverse coupon collector problem

Question

So in coupon collector, we calculated the purchases needed to get all coupons.

I wonder how to calculate the expected number of coupons seen given we made $k$ purchases.

Formally, we have points {1,2,3, $\cdots$, n}. Each point has probability $1/n$ to be drawn. If we make $m$ draws, and put each draw into a set $K$. What is the expected size of the set $K$?

Answer 1

Hint:

• What is the probability that item $1$ is drawn in one attempt?
• What is the probability that item $1$ is not drawn in one attempt?
• What is the probability that item $1$ is not drawn in $m$ attempts?
• What is the probability that item $1$ is drawn at least once in $m$ attempts?
• What is the expected number of distinct items drawn at least once in $m$ attempts?