This is the first of a regular set of puzzles in Combinatorics, Algebra, Probability, and Statistics (CAPS). Simulation will always get you a solution. But my aim and hope is to make these as accessible classroom activities either acting them out physically or working out the math. The end goal, of course, is to start a conversation and get us thinking about topics beyond the textbook, but still working with the math that is available in the textbook!

RULES

Take a six-sided die and keep rolling it until a six is rolled. Once a six is rolled the game is over. In the meanwhile, keep two sets of counts: the first count (call it “A”) is if the die roll is 5 or less. The second count (call it “B”) is if the die roll is 4 or less.

What is the probability that A > B?

or put another way:

What’s the probability that there will be at least one 5 in the sequence of rolls before a 6 is rolled?

This may sound confusing, so here’s an example of how the dice rolls and counts work. Sample dice rolls: 3,4,5,4,2,1,1,6. The 6 terminates the game. The A count is 7 since there are seven dice rolls less than or equal to 5. The B count is 6 since there are six rolls less than or equal to 4.

Here’s another example: 4,2,1,6. The A and B counts are 3 because there are three rolls that are less than or equal to 5 (A count) and three rolls that are less than or equal to 4 (B count).

This should be solvable by hand! Let me know if you want a hint or if you have the answer!