Finishing up our project on Pasquale Cirillo’s introductory probability / expected value problem

Yesterday my younger son and I talked through this fun problem I learned from Pasquale Cirillo:

Our project is here:

https://mikesmathpage.wordpress.com/2021/05/01/talking-through-a-neat-introductory-probability-expected-value-problem-from-pasquale-cirillo-with-my-younger-son/

and Cirillo’s discussion of the problem is here:

Brain Teasers for Quants, Episode 9: A simple dice game and American options in disguise. https://t.co/CXP3HNv8b3

— Pasquale Cirillo (@DrCirillo) April 26, 2021

Yesterday we did not get to the optimal solution, but rather looked at the strategy of stopping when you get a 6 on the first or second roll, and then at stopping when you get a 4 or higher on the first or second roll. I asked my son to think about the problem a bit more this morning while I was out and he was able to find the optimal solution.

Here’s what he did in his own words:

Next he showed how he used Mathematica to help him find the best solution:

Finally, I showed him an alternation approach to finding the optimal solution that comes from working backwards. This is the approach that Cirillo takes in his discussion of the problem: