In 1818, the French mathematician Simeon-Denis Poisson showed that if the wave theory of light (the view that light is a wave propagating in a medium) is correct, then if a steel ball is placed between a point light source and a screen, a very bright spot of light will appear at the center of the circular shadow of the ball on the screen. Poisson was a supporter of the corpuscular theory of light (the view that light is not a wave but a stream of corpuscles) and thought that this apparently absurd prediction would deal a death blow to the wave theory. But the French Academy of Sciences arranged for an experiment, and the bright spot was actually observed exactly where the wave theory predicted it would be. This was rightly taken as a dramatic confirmation of the wave theory. Let's try to understand why.
Let's call the wave theory of light T, and the prediction that that a very bright spot of light will appear at the center of the circular shadow of the ball on the screen, E. Let's also assume that 0<Pr(T)<1 (i.e., we consider T neither patently false nor conclusively established since we are testing it), and 0<Pr(E)<1. Then Poisson showed (this is a bit of oversimplification) that "T; hence E" is a deductively valid argument. Of course, he thought that E was false, and he inferred that therefore T was false as well. His reasoning was correct: if an argument is valid and the conclusion is false, then the premise(s) cannot be true. Unfortunately for him, however, E is not false but true. Does this show that T is true? Not strictly, since a valid argument with a true conclusion can have false premise(s). Still, E's being true should increase our degree of belief in T if Pr(T|E)>Pr(T), that is [Pr(T|E)]/[Pr(T)] > 1.
Now, by Bayes' Theorem, Pr(T|E)=Pr(E|T)x[PrT)/[Pr(E)].
But since 'T; hence E" is valid, Pr(E|T)=1. Hence, Pr(T|E)=[Pr(T)]/[Pr(E)]. So, [Pr(T|E)]/[Pr(T)]=1/Pr(E). But since 0<Pr(E)<1, it follows that [Pr(T|E)]/[Pr(T)]>1, and therefore E corroborates T.
This is why if a theory T passes a test, our degree of belief in T increases. Notice also that the smaller Pr(E) is, the greater the degree of confirmation; that is, the more surprising the corroborating phenomenon, the greater the corroboration of the theory that predicts it.