During our holiday celebration, my family started a game of zilch. Zilch is a dice game played with six dice, and is somewhat similar to yahtzee. I'm interested in the probability of getting a zilch in the first roll, with the six dice.
In the first roll, the following situations score:
any die 1
any die 5
three or more of a kind
If you fail to get any of those, you score 0 and lose your turn (you "zilch"). Assuming fair dice, what is the probability of zilching?
Bonus: What about with 5, 4, 3, 2, or 1 die? The same scoring rules apply, except obviously you can no longer get a straight or three pair, nor can you get a triple with 1 or 2 dice.
[In the game, you set aside at least one die that earns you points from each roll and have the option of rolling the remaining dice. You add all points you earn to your score; however, if you zilch at any point in your turn, you get no points whatsoever for that turn. So knowing how likely it is to zilch is fairly important. 1 and 2 dice are easy, 3, 4, 5 a bit harder, and I think all 6 is a bit tricky. I have an answer for 6, but want to make sure I've got it right.]