Test Your ESP!
My sister once bought me a book entitled Test Your ESP. This book contained some standard tests for ESP. For example, someone might deal a card face-down from a well-shuffled deck containing equal numbers of five different kinds of cards. You attempt to determine what sort of card it is without looking at the face, and then check to see whether you were correct, and then repeated this procedure over a number of trials. But the book had an interesting twist concerning the interpretation of the results: if you scored better than chance, you probably had ESP, for obvious reasons; if you scored worse than chance, you probably had ESP, and were apparently unconsciously attempting to get the answers wrong, perhaps to conceal your abilities; and if you scored exactly what chance would predict, you probably had ESP, because scoring exactly what chance would predict is actually very unlikely. One way to look at this is to say that the theory predicted that if you had ESP you would score either above chance, or below chance, or at chance. But of course these are the only three possibilities!
So now we check to see whether this prediction comes true. Lo and behold, it does. What should the posterior probability of the hypothesis that you have ESP be, following the book's procedure? Well, let's say that you are initially agnostic again, so P(T) = P(~T) = .5. The theory (plus background information) predicts the observed result with certainty, so P(E|T) = 1. On the other hand, the observed result would have been obtained in any case, since it is logically necessary, so P(E|~T) = 1. What happens if you plug in these values? Here's the applet again to try it out:
You might also consider the following cases in which a theory "predicts" something that is likely to happen in any case. Theory: you can cure the common cold by some holistic remedy. Evidence: a few days after taking the remedy, the cold improved. Theory: birth control pills prevent pregnancy in men. Evidence: Mr. Jones, who has been taking birth control pills, has not become pregnant.