I was fortunate enough to attend the Columbus Math Teacher Circle yesterday and to share the game Borel as a way to introduce my favorite instructional arc for teaching probability ideas: Intuition, Simulation, Calculation.

The first problem we looked at was this:

Roll three 6-sided dice and one 30-sided dice. Will the difference between the maximum value and minimum value showing on the dice be greater than 15?

Pause for a moment and think about this problem. What does your gut tell you? Then roll these digital Polypad dice to get a feel.

Thinking a little about this scenario...

If you roll a 16 or less on the 30-sided die, there is no way to get a max–min difference greater than 15. So the only possible way to get a difference greater than 15 is to roll a 17 or greater on the 30d. That is 46.6% of the rolls. But for the roll (17,2,3,4) the difference is 15, so not

*every*roll will produce a difference greater than 15. So the probability of this happening is going to be less than 46.6%.Here is a GeoGebra simulation as well, where you can run 1 trial or 10 trials at a time, and see a summary of all the max–min differences.

Here is a python script that looks at every roll and and keeps track of the differences

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