I'm doing a little self study in probability theory and read about the coupon collector's problem, which lead me to think about the following dice game:
Suppose you have seven regular fair 6-sided dice, labeled $d_0, ..., d_6$. In each round of the game first $d_0$ is rolled. Let's say it lands on $i$; then $d_i$ is rolled.
I'm interested in the number of rounds until all six sides of all seven dice have been rolled (at least once). How would I go about analyzing this situation? In particular, how can I find the distribution of number of rounds and/or what is the expected value?
What about the variation where I'm just interested in rolling a particular side (say, 6) of all seven dice? And what if we generalize to a $k$-sided "selector" die that chooses which $n$-sided die to roll? As you can probably tell, I'm completely stuck on these problems.
For a single die the situation is easier; it's just the coupon collector's problem, or for the mentioned variation, the number of Bernoulli trials needed for one success with $p = 1/6$. But as soon as I introduce a second die (and a coin flip to choose between them), the problem seems to become much more complicated, let alone the situation described above.
I've tried to read up on probability generating functions, but so far I haven't been able to tackle the problem. I've also tried modeling the problem as a Markov process, but that seems intractable for dice with more sides.