There is a healthy number of turn-based dice games with a press-your-luck element such as 10,000, Farkle, and Greed. Here is a variant that keeps the numbers smaller and probabilities simpler in order to make the arithmetic and analysis more tractable for younger audiences.
In this activity, students design bingo cards based on their impressions of which sums will come up first and most often when two dice are rolled repeatedly. This activity is a nice introduction to discrete probability distributions and some of their conceptual pitfalls.