Yes, I have designed a game with dice. Yeah, I know. Sad panda. Anyway, my revulsion at this has meant I have to be extremely careful about how these entropy generators might behave as the game is played. I'm prepared for long-tail scenarios. This is (Alanis Morrissette) ironic because FCOJ is a game about how people failed to prepare for long-tail scenarios.
Anyway, the chart below is a breakdown of how many hits (at least) on these dice will occur with 3, 6, 9, and 12 dice rolled. The game is divided into 3 rounds and at the end of each round 3 dice are rolled (plus more if certain things have happened). The hits are cumulative over the course of the game, so it that's why I structured the visualization that way. It allows me to see how likely a cumulative total is to happen as well as when during the game is it likely to happen. This is key to making sure the game has an arc which affects player emotions at just the right points.