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I'm working on a problem where I need to find the expected number of throws required for each face of a six-sided die to appear at least twice. I've conceptualized the problem using a triplet $(a, b, c)$ to represent the state of the system:

  • $a$: numbers drawn twice,
  • $b$: numbers drawn once,
  • $c$: numbers not drawn,

with the initial state being $(0, 0, 6)$. The transitions are:

  • From $c$ to $b$ with probability $\frac{c}{6}$,
  • From $b$ to $a$ with probability $\frac{b}{6}$.

And then you can use brute force and write an algorithm to solve it using DP. But I’m wondering if there is any clever method. I know a lot of theorems have been studied on coupon collections issues. But this is a particular simple example so I’m curious about some simpler but clever strategy free of those existing theories that are intended for a more general scenario.

joriki
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The One
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  • I changed the duplication notice to point to questions that have answers with the general result for this problem, which is called the “double Dixie cup” problem. One has the specific result for obtaining $2$ copies of each coupon, the other the general result for any number of copies. In your case with $n=6$ coupons and $m=2$ copies, the result is

    $$ 6\int_0^\infty\left[1-\left[1-(1+x)\mathrm e^{-x}\right]^6\right]\mathrm dx=\frac{390968681}{16200000}\approx24.134;. $$

     – joriki Feb 25 '24 at 13:24
  • @joriki is there any clever way for this reduced case of m=2? For m=1 for example you can just use linearity of ev and get 14.7 by mental math. It seems for m=2 you have to take integral or do some algorithm instead of computing by hands – The One Feb 25 '24 at 23:35
  • I’m not aware of any simpler calculation than this. The integral can be expressed a sum, but I doubt that you’d be able to evaluate that sum by “mental math” :-). You can also look at the answer to the question that was originally linked to in the duplication notice here; your case is $n'=0$ and $n=6$, but that double sum is also rather complicated. – joriki Feb 26 '24 at 01:29

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