What is evidence?

Some people define evidence like this:

E is evidence of T just in case the probability of T is higher given E than it would be without E.

Or in other words. . .

P(T/E) > P(T/~E)

I once used that definition of "evidence" in a debate to show that contrary to my opponent's view that "faith is not evidence" for the existence of God, faith actually was evidence for the existence of God since the existence of God is more probable given that some people have faith than it would be if nobody had faith.

I'm not sure that's really a good definition of evidence, though. Consider the existence of the Elder wand in Harry Potter. The Elder wand is a magic wand made from the Elder tree that is so powerful, it renders the wizard who wields it invincible. That's assuming the wand chose the wizards since in Harry Potter, the wand chooses the wizard.

Now obviously if there were no such thing as an Elder tree, then the probability of the existence of the Elder wand would be zero. But the Elder wand would at least be possible if the elder tree existed, which it does. That possibility might be extremely remote, but the probability would still be greater than zero. Something whose existence is possible has a better chance of existing than something whose existence is impossible, so the probability of the existence of the Elder wand is greater given the existence of the elder tree than it would be given the non-existence of the elder tree. From that, it should follow that the existence of the elder tree is evidence for the existence of the elder wand.

But doesn't that strike you as wrong?