There's an interesting ("interesting if you're a boring person!", which I am) analogy to those situations you sometimes see in TV poker, and quite possibly in real-life ftf poker, where you have folded to a villain who you believe to have a specific pocket pair. Villain offers to show you one of his cards and it is indeed of the specific rank. These are more probably questions of villain's psychology (and choice of card revelation protocol) than of probability, but:

1) Is the chance that he has the pocket pair different if you've chosen which of the two cards to see to the chance that he has the pocket pair if he's chosen which of the two cards to show you?

2) Is the chance that he has the pocket pair different if it's your idea to pick which card to see than if it's his idea to pick which card to see?