Problem #9 from Mosteller’s 50 Challenging Problems in Probability is about the game of craps. The question asks, essentially, does the player of the casino have a better chance of winning the game.
This is both a fun and reasonably difficult problem for kids, but it led to a terrific conversation.
Here’s how I introduced the problem:
Following the introduction, I had the boys solve for the probabilities of an immediate win or loss:
Now we moved on to the harder question – what happens if you roll. say, and 8 on the first roll. How do we find the probability that you win the game in this situation?
One way would be summing an infinite series, but I hoped to introduce the boys to a simpler way of seeing the probability here:
Having solved one of the hard cases exactly in the last video, we moved on to solve the rest of them here:
Finally, we went to Mathematica – not for anything super complicated, just to add up the fractions – so we could find out whether or not the player or the casino had the advantage in the game:
This problem is a great one for kids to explore – it really shows how a systematic approach to problem solving can help you get throw a pretty challenging problem.