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Question:
Say I have a set of 24 dice that have different letters on them, what is the chance that in any roll a certain combination of letters appears at least once?

Combination I want:
{a, a, b, c, d, e}
(a is duplicated on purpose)

Dice:
The 24 dice would come from a pool of 4 different colors, with each color having different possible letters
6 red dice: {a, b ,c, z, z, z}
6 blue dice: {a, b, d, z, z, z}
6 green dice: {d, e, z, z, z, z}
6 black dice: {c, z, z, z, z, z}
(z represents undesirable rolls to fill out the 6 sides of the die)

Problem:
I really don't even know how to start on this. I've looked up other questions about probability and set theory, but I haven't found anything on dice with repeating letters or on sets with multiple duplicate elements.

As an aside, I'd eventually like to implement the solution in python so I can check many subsets at once, so any answers put in coding language would be heavily appreciated.

  • Could you please explain what "contained in those dice" means and why color has any importance in this question? Are the dice being rolled or are the values of "z" being chosen randomly or something else? – whuber Apr 03 '19 at 22:10
  • @whuber Colors are just a way to differentiate between the dice with different letters on them. contained in these dice means that all of the letters in my desired set show up at least once when all 24 dice are rolled. z just represents letters on the dice that I don't care about that are also possible rolls. –  Apr 03 '19 at 23:34
  • One general approach is described at https://stats.stackexchange.com/questions/124739. Others of this nature can be found by searching [polynomial ideal modulo](https://stats.stackexchange.com/search?q=polynomial+ideal+modulo). – whuber Apr 04 '19 at 12:48

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