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Here is the problem:

  • You have N dice of size D. You throw all dices, then for each dice:

    • if a dice face is d in D leave it like that
    • otherwise re-roll the dice, until you get d

  • What is the expected probability distribution (is this a correct term?) for a total number of individual dice throws to get d on all dices ?

My question is:

  • Is there a name for such problem ? Or a name of a branch of probability theory to deal with this ?

Similar question: Let $N_n$ be the number of throws before all $n$ dice have shown $6$. Set $m_n := E[N_n]$. Write a recursive formula for $m_n$. has answers, but does not explain what are we searching for.

joriki
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c69
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  • This is kind of "jeopardy" question ("guess the question given answer"), but I really don't have the mathematical education to formulate it on my own, and intuition tells that this should be something that students study on their first or second year. – c69 Aug 18 '18 at 02:47
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    I am not sure you would expect this question to be in a class which has a name, but the approaches are similar to those you would use where the underlying distributions were geometric or negative binomial or binomial, so discrete distributions on non-negative integers. For example, the CDF (i,e, the probability of needing $n$ or fewer throws) is $\left(1-\left(1-\frac{1}{D}\right)^{n}\right)^{N}$ – Henry Aug 18 '18 at 21:18

1 Answers1

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  • all dice are the same (and independent), so calculate for 1, then multiply by N
  • all faces are equal (that’s a definition of a die, right?), so d=1

So a simplified problem is: “number of rolls until getting 1 on it”. Which is a geometric distribution with $p=1/D$

The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}

kirilloid
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