Name the problem: Numbers of dice throws to get result with rerolls allowed

Question

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.

Answer 1

• 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, ...}