So who won the debate?

A friend of mine got his driver’s license today. He was worried that he may not pass the driver’s test, but I kept saying he would. So his passing the test gives me a perfect opportunity to go all “I told you so!” on him. But mathematically speaking, am I justified in doing that?

Things are much easier in a deterministic world, or even in a world where all our wagers were deterministic. So let’s talk about that world for a while. Suppose your friend says he will definitely fail a test and you say he will definitely pass the test. Then it is very clear who won the debate once you know the outcome of the test. Of course, you win if your friend passes, he wins if he fails.

But the friend in question is mathematically more sophisticated. When I told him there was no need to worry and that he was going to pass the test, he didn’t say he was definitely going to fail. He said that there was a greater than 25% chance that he was going to fail.

Let’s assume, for simplicity, that I’d claimed his passing to be an absolute certainty. His claim estimated the probability of passing to a modest 75%. Now given that he did pass, who won this debate?

The answer is that it’s complicated. We can’t say that I won, because perhaps the true probability of his passing was indeed 75%, and this specific instance of the test happened to be drawn from the 75% of the instances where he does pass. Can we say that I lost? No, because perhaps the true probability was actually 100%.

The real answer is that in the middle of all these probabilities, we should not expect to have a definitive winner of the debate. Rather, all we should expect to extract from this event is a probability that I was the winner. A mathematically correct arbiter will start with an impartial prior probability about who’s the winner and use the outcome of the test to merely update this probability using the Bayes theorem.

I’m going to meet this friend in about 20 mins. Mathematically speaking, am I justified in saying, “I told you so!”? No. But am I going to do it? Yes.