Thinking about the lottery

I feel as though I ought to write something thoughtful about the lottery, since the Mega Millions draw has got to be an impressively high value, and there are fine things to be said about probability, expectation values, and what we really mean by the probability of an event when we can only take one attempt at doing it (since I can’t make myself think the lottery will go another week without a winner and it is likely to be years before such a big jackpot rises again).

And in fact I did go into it in some detail in the course which has got into probability today. There it makes an inevitable fit, since we were at the subject of expected values — what the average result of an experiment, where the standard examples are dice throws or very simple games of chance, will be — anyway and the nebulous figure of a half-billion dollars is a great way to make this stuff sound more immediate. Most of my students would be happy even with the quarter-billion projected for after taxes under some conditions that the TV news probably (sensibly) didn’t go much into.

Expectation values are a peculiar subject; the idea behind them is almost too simple to teach — I could see students with that faintly baffled look which signifies, ”this is too easy; where’s the real, hard, stuff?” and there just isn’t — and it’s almost universally applied, though to always by name. But whenever one is deciding whether it’s better to repair the old car with its strangled transmission or replace it with a different used car with new set of defects or buy a new car one is working out an expectation value, of what car-based costs are going to be. When someone has to work out the cost-effectiveness of a proposed plan, that’s again the expectation value creeping in.

But for many of the cases where we rely on the expectation value we don’t have the information we need to make a decision. In the question of fixing or replacing the car, we can make guess at the costs of repairs for the next year (or whatever), or the likely costs of a replacement used or new far, but figuring out the likelihood of possible needed repairs is something only a few have. It’s odd to know there is a tool which should be able to provide rigid guidance based on a single value (the money keeping or replacing this car would cost; I am not ignoring that cost is far from the only factor that goes into car decisions) and yet can’t be used in more than a loosely approximate way.