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Say we have a set of $n$ values. At each attempt, one of these can be obtained randomly, with equal probability ($1/n$). The set doesn't change, so a previously obtained value can be gotten multiple times repeatedly. The goal is eventually obtaining all distinct values from the set.

Someone "with minimal luck" would get the same value forever and never reach the goal.
Someone "with maximal luck" would reach the goal in the minimum of $n$ attempts.

Questions (generic):
  1. How many attempts $a$ should someone "with average luck" expect to need until all values are discovered?
    What does the plot of this function $a(n)$ look like?
  2. How would this person's set of distinct values obtained $c$ be expected to grow with the number of attempts $a$ for a certain $n$?
    What does the plot of this function $c(n,a)$ look like (e.g. for $n=25$)?
  3. What's the probability $p$ that this person would eventually reach the goal, given a maximum of $a_M$ attempts?
    What does the plot of this function $p(n,a_M)$ look like (e.g. for $n=25$)?
    What does the plot of this function $p(n,a_M)$ look like (e.g. for $a_M=90$)?
  4. Is this some sort of named "classical problem" in statistics? What should someone search for to get some of these formulas?
    Or alternatively, how do you formulate them?

Plots are optional, I can get those in Wolfram Alpha if I know the formula. But they would make a better answer.

Same questions, as an example specific problem:

A certain brand of cookies is running a promotion this year, where they include a random collectible card inside every package. These are known to be a total of 25 cards randomly distributed with equal probability (no, really!). The promotion lasts 90 days. Assume everyone has average luck.

  1. If an interested collector wants to complete the whole set, how many cookies should he expect to need to buy (if he can't trade any cards)?
  2. What function describes how his collection of distinct cards is expected to grow with every purchase? What does it look like?
  3. A kid (not me!) has enough money to buy 1 cookie every day, but he has no friends (again, not me!) that can trade cards with him. How likely was I is he to complete the entire collection before the promotion ends?
RobPratt
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geekley
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    Unsurprisingly, this is popularly known as the Coupon Collector's Problem. – lulu May 11 '23 at 23:25
  • I tried searching, and I knew this had to be some sort of classical problem, but I didn't know what to look for or how to phrase it. Thanks, @lulu now I know I can answer at least (1) and (4). My intention is to help anyone with such question find the name of the classical problem. – geekley May 11 '23 at 23:45
  • Your questions 2 can be calculated as $n$ times the probability value/card/coupon $1$ is found at least once in $a$ attempts. Your questions 3 can be calculated using inclusion-exclusion or Stirling numbers of the second kind. Both have been asked several times here – Henry May 12 '23 at 01:35
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    See for example https://math.stackexchange.com/questions/28905/expected-time-to-roll-all-1-through-6-on-a-die and https://math.stackexchange.com/questions/3441301/inverse-coupon-collector-problem and https://math.stackexchange.com/questions/2499326/roll-an-n-sided-die-m-times-what-is-the-probability-that-1-2-3-n-faces – Henry May 12 '23 at 01:41

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