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James Fung
United States
Berkeley
California
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Lately, I’ve been thinking about how I would teach various classes I may be called to teach. Frankly, I’m uncertain how to design a project-based course on, say, linear algebra or calculus. But for probability and statistics, I have some ideas: games. There are 2 reasons I want to use games: 1) Games are fun, which I hope will engage students. I want them to want to learn more, to explore a problem and try to solve it for themselves, something I find lacking in many math students. 2) I like games and can talk about them all day. I hope that will give my lessons that little extra fizz since I’m not the most charismatic teacher.
One excellent game that comes to mind is Rallyman. I don’t have time to summarize the rules, but it is an excellent mix of risk management and planning. Here are some ways the game can be used to illustrate various concepts:
Intro. If people don’t know what rally driving is, show them a video of Ken Block or something. Will this turn off people who aren’t interested in cars? Maybe. Have to work on that.
Probability and events. Start with a time attack roll, when all dice are rolled together. What is the probability that 3 hazard symbols come up and you lose control? Well, that depends on the mix of dice being rolled. Start with simple problems of all dice having the same probability of hazards (1-in-6 or 2-in-6), then move onto mixtures. This can lead to discussion of counting (how many ways are there to roll 2 hazards with 4 dice?) and/or making use of the probabilities of the events they just calculated.
Another concept that’s very important here is independent events: the result of one dice doesn’t affect the other dice. Furthermore, say you’re rolling 1 gas and 3 gear dice. We can create events A, one hazard is rolled on the gas die; B, 2 or more hazards are rolled on the gear dice; C, 3 hazards are rolled on the gear dice. Event A is independent of B and C, which makes it easier to calculate probabilities of intersections of events.
Conditional probability. I’m pretty sure there’s an example of conditional probability, but a lot of the simple events (i.e. die rolls) are independent, and the only example I can think of (how one turn impacts the ones after it) is probably better done after discussing expectation.
Random variables and expectation. The bottom line in Rallyman are seconds spent, which is an example of a random variable: if we make this move, we have an 80% chance of using only 10 seconds but a 20% chance of using a whole minute. From this, we can talk about expectation: on average, how many seconds will this take? (Of course, maybe taking only 10 seconds will win you the race while 60 seconds will lose it, so depending on the context, asking about expected number of seconds is the wrong question, but that will have to wait until we talk about cost functions and decision theory.) From there, we can extend this to conditional expectations: if you spin out, that screws up your next turn, which affects the number of seconds that will take.
Big picture. I don’t expect Rallyman to be solved by any student without a computer. I think if Rallyman were used in a classroom, its role would be to illustrate various concepts and to show sample calculations. For evaluation, students would instead use these ideas to study something else.
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