A short talk about probability and game theory

At least that’s what I thought – it ended up being another good example in the “what learning math sometimes looks like” series.

Last night I saw this book out of the bookshelf.

I don’t think anyone was reading it, we just had a few different people staying with us for the holidays and had to move a few things around. Seeing the book made me think of using something from it for our Family Math project this morning. I ended up finding two good ones.

The first is a problem about the archers shooting at a target. You are told the probability for each one of the archers hitting the target on a given shot and then asked to calculate the probability that at least one person hits the target if each of them takes one shot.

A fairly standard probability problem but we haven’t talked about probability in a long time, so I didn’t quite know what to expect. During the first part of the talk the kids were pretty confused. We don’t make a lot of progress towards the solution and approaching the 5 minute mark my younger son has the idea of adding up the individual probabilities. We’ll explore doing that in the next video:

I decided to stop the last video after 5 minutes just to split the conversation into two pieces. All I did was turn the camera off and on and we picked up where the last movie left off: talking about adding up the probabilities. Eventually our little winding path takes us to the idea of looking at the probability that 0 people hit the target in a round. The idea of looking at the times when 0 people hit the target – a concept called complementary counting – is an important (and not obvious) idea in counting and probability. Our conversation about complimentary counting here is far from complete!

The next problem is involves a little more probability and a little game theory. I think this is the first time that we’ve ever talked about game theory, so I was really excited to see how the kids would react to this problem. The problem goes like this:

Three people play a dodge ball-like game. Each person takes a turn throwing a ball at an opponent. If you hit an opponent that person is out, and the goal of the game is to be the last person standing. Two of the three people get hits 100% of the time and the third person gets a hit only 50% of the time. Who is most likely to win this game?

Fun little problem with a solution that isn’t at all obvious. I find problems like this to be more difficult to think through than the first problem, but the kids seemed to find this problem to be much easier to understand. It was fun to hear them think through the different ideas and get to the surprising result:

So, a fun morning with counting and probability (and a little game theory!). It has been a while since we talked about these topics and the first problem gave the kids quite a bit of trouble. The lesson there for me is that it probably makes sense to return to old topics occasionally rather than just leaving them for good. The game theory problem turned out to be more fun than I was thinking it would be. Hopefully I can find some more fun game theory problems are accessible to kids.

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Comments

2 Comments so far.
1. Love the game theory question. You might recall in the 2012 Olympics there was a scandal involving badminton teams intentionally losing (or trying to lose) that has parallels to the simple dodge-ball game.

One other example from game theory that is really interesting is where a board of directors votes on a CEO from a slate of 3 candidates. Given suitable preferences, you can create a scenario where the order of voting can be set to allow any of three candidates to win. Though obviously stylized, it is an important illustration that process matters (and controlling the process is powerful). If corporate politics won’t resonate with your sons, could do the same thing with choosing a restaurant for a family dinner or vacation destination.

It also has resonance with transitive dice. I assume your kids know about that, but would probably enjoy if not.

• we have not talked about transitive dice, but that is an absolutely great idea. I’ve got a little bit of travel in the next few weeks, but look for that Family Math coming soon!

On the voting – I really enjoyed the section in Ellenberg’s “How not to be Wrong” on voting theories. I mainly studied mathematical physics and differential equations, so stuff like that is pretty far outside of what I learned in school.