The Universe of Discourse

Wed, 22 Nov 2017

I think this example is very illuminating of something, although I'm not sure yet what.

Suppose you are making a short journey somewhere. You leave two minutes later than planned. How does this affect your expected arrival time? All other things being equal, you should expect to arrive two minutes later than planned. If you're walking or driving, it will probably be pretty close to two minutes no matter what happens.

Now suppose the major part of your journey involves a train that runs every hour, and you don't know just what the schedule is. Now how does your two minutes late departure affect your expected arrival time?

The expected arrival time is still two minutes later than planned. But it is not uniformly distributed. With probability !!\frac{58}{60}!!, you catch the train you planned to take. You are unaffected by your late departure, and arrive at the same time. But with probability !!\frac{2}{60}!! you miss that train and have to take the next one, arriving an hour later than you planned. The expected amount of lateness is

$$0 \text{ minutes}·\frac{58}{60} + 60 \text{ minutes}·\frac{2}{60} = 2 \text{ minutes}$$

the same as before.

[ Addendum: Richard Soderberg points out that one thing illuminated by this example is that the mathematics fails to capture the emotional pain of missing the train. Going in a slightly different direction, I would add that the expected value reduces a complex situation to a single number, and so must necessarily throw out a lot of important information. I discussed this here a while back in connection with lottery tickets.

But also I think this failure of the expected value is also a benefit: it does capture something interesting about the situation that might not have been apparent before: Considering the two minutes as a time investment, there is a sense in which the cost is knowable; it costs exactly two minutes. Yes, there is a chance that you will be hit by a truck that you would not have encountered had you left on time. But this is exactly offset by the hypothetical truck that passed by harmlessly two minutes before you arrived on the scene but which would have hit you had you left on time. ]