Yesterday we studied problem #7 from Mosteller’s 50 Challenging Problems in Probability. That project is here:
The problem is about a game of roulette and has the surprising result that you have a better than 50% chance of being ahead after 36 bets. Last night I realized that the project might have accidentally left the kids with the impression that the game had a positive expected value – whoops!
So, today I wanted to be sure that they did not have this impression. In thinking about how to talk through this topic last night, I realized that some of the ideas that Ole Peters has shared recently are somewhat similar, so I decided to share those ideas with the kids today, too.
We started by reviewing the results of yesterday’s project:
In the initial conversation the boys thought that we should look at how much you won when you were ahead and how much you lost when you were behind. Off camera we modified our program from yesterday to address these questions.
Here we talk about the program and then see the results.
Now I introduced the boys to Ole Peters’ coin flipping game from the talk below. We watched the 5 min segment from roughly 4:00 to 9:00 where Peters explains the game and shows a pretty surprising result:
I’d originally intended to play around with a computer program to simulate Peters’ game, but we were running a bit long so I decided to just talk through it.
The boys were a little surprised by the results, but I think they were able to understand why the outcome for the individuals was different from the outcome for the group.
I really enjoyed this project with the kids today. Hopefully the two simple, but somewhat surprising, ideas from today stick with them:
(i) Having a high chance of winning doesn’t mean a bet is a good bet, and
(ii) Even if the result of a game is positive for a large group, it can still have a negative outcome for almost everyone in that group.