Help calculating probability of dice rolls in a board game

by Cassie Thompson   Last Updated November 09, 2018 00:20 AM

A long time ago I studied math (even taught it!) but I have forgotten it all. I was playing a board game with my kids and the dice in the game really got me wanting to know what the chances of success were.

I tried to work it out on my own, but I have been out of the game too long, I would love some help:

  • The game has eight identical six-sided dice. Each dice has five symbols, the sixth symbol is a duplicate.
  • The player is allowed a one-time reroll up to seven of her dice.
  • The goal is to match a six-symbol combination.

To be honest, I don't know where to begin. What follows is going to be sketchy swinging in the dark.

The number of possible combinations that could be made with n=5 symbols and r=6 spots is 210.

i.e. $$\frac{(r+n-1)!}{r!(n-1)!} = \frac{10!}{6!4!} = 210$$

But that has nothing to do with the actual dice.

I figure I can calculate the rerolls by considering the dice as 15 dice being rolled, not just 8. So I am after the possibility that 6 of the symbols out of the 15 dice rolled, would match the 6 symbol combination that I am looking for.

(For the record, the game kind of sucks, and I think it is because of the probability of success. I have serious doubt as to whether the game designed looked this far into it either!!)

Example of a symbol combination

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