Coupon Collector Problem solving for expected completion percentage

Question

I'm working on a variation of the coupon collector problem and trying to make a formulaic solution in Excel without using a user-defined function.

Basically, the gist of the problem is that at a variable number of attempts (t), and I already have a number of coupons collected (n) of the total coupons (x), I'm trying to solve for expected completion percentage (P.) The formula needs to be able to handle different values of t, n, and x,

In short:

• I have t attempts to pull my coupons
• I have n coupons already collected
• There are x coupons in total
• Knowing those variables, I'm solving for P - expected completion percentage

What is the formula to define expected completion percentage?

P = ?

EDIT: Basically, the coupon collector problem solution I have is this:

Normal coupon collector solution:

p = n$\sum\limits_{k=1}^n$ $\frac 1k$

EDIT2:

Simply phrased, I want x-n coupons. I have t tries to get those coupons. What is my expected completion percentage after those t tries?

Answer 1

For each of the $x-n$ coupons you do not already have:

• the probability of drawing it on the next attempt is $\frac{1}{n}$
• the probability of not drawing it on the next attempt is $1 - \frac{1}{n}$
• the probability of not drawing it on any of the next $t$ attempts is $\left(1 - \frac{1}{n}\right)^t$

So overall, since expectation is linear, after $t$ attempts

• the total number you expect not to draw is $(x-n)\left(1 - \frac{1}{n}\right)^t$
• the total number you expect to have (including the starting $n$) is $x-(x-n)\left(1 - \frac{1}{n}\right)^t$
• the total proportion number you expect to have is $\dfrac{x-(x-n)\left(1 - \frac{1}{n}\right)^t}{x}$