I've been struggling with this for a while:
Info: I have a bot on Discord that posts a random question every 4.5 minutes (on average). It posts the question from a database I prepared, which has 150 unique questions. Questions can repeat.
I want to find the probability that all the questions were posted atleast once, in $150 + n$ posts (i.e in $675 + 4.5n$ minutes). I know the probability as:
$$p = \frac{\text{possibilities of getting all questions atleast once}}{\text{all possibilities}}$$
For the denominator, it's simply $150^{150+n}$.
For the numerator, for some 150 questions it posts, we have $150!$ ways of getting all questions atleast once, and for the remaining $n$ questions, we would have $150^n$ ways, giving us $150! \times 150^n$.
Piecing this together, I should get:
$$p = \frac{150! \times 150^n}{150^{150+n}}$$
$$p = \frac{150! \times 150^n}{150^{150} \times 150^n}$$
$$p = \frac{150!}{150^{150}}$$
This makes no sense. It says that the probability is independent of $n$, which is completely absurd, since with increasing $n$, the probability that you'd see all the challenges should increase!
What have I done wrong?
This is the coupon collector's problem. The probability distribution you're looking for is given at Probability distribution in the coupon collector's problem.
Your count is wrong because you're undercounting by not taking into account that different sets of $150$ posts can contain all questions (there are $\binom{150+n}{150}$ such sets).