(Not So Typical) Dice Problems

Probability and counting questions associated with rolling two dice are frequent at math contests. For example, one might ask, "What is the most common sum when two dice are rolled?" The grid below helps the student see the most common sum.

Exhibit A: The sums of two dice. (Ignore the pink stuff; 7 is the most common sum).

Suppose we construct the probability distribution for rolling two dice.

Exhibit B: Probability simulation for rolling two dice. 7 is the most likely sum.

One of my colleagues has 11 dice in her classroom. This made me wonder what would be the most likely sum for 11 dice. How would we go about answering this question?

Problems like this one show students the value of breaking a larger problem into smaller problems. Let's construct a table for the minimum and maximum sums for rolling n number of dice and see if we can determine some patterns. Our aim is to generalize our findings to n dice.

Exhibit C: A preliminary table for rolling n dice.

How would you support students in working towards the generalization? For example, modeling two dice with a table is fairly simple, as shown above. What about modeling three dice? Sure, you could go three dimensional... but what about four dice? Five dice? This problem poses some great modeling questions.

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Exhibit D: Polynomial approach to obtaining dice counts for the case where n = 2.

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