Two Ace Problem

From Qwiki

I deal you two cards from a standard 52 card deck, I look at both of the cards. I make a true statement about your hand. The allowed statement depends on what game we are playing. In game 1 the allowed statement is:

"You are holding at least one ace"

while in game 2 the allowed statement is:

"You hold the ace of spades" (for hands where you actually do hold the aces of spades.)

Now the strange thing is that the probability that you hold two aces, given that I've just said something, depends on which game we are playing. In particular, a statement made during game 2 carrier almost twice as much weight as a statement in game 1. That is, if we're playing game 2, and I say "You hold the ace of spades," then you are almost twice as likely to be holding two aces then for the case where we're playing game 1 and I just said "You are holding at least one ace." Couldn't you have simply pretended that your ace was the ace of spades (or the ace of hearts for that matter)?

Now you can calculate all these probabilities fairly easily using counting arguments, conditional probabilities, etc... All the answers are consistent, of course, but getting comfortable with the results can take some time.

The Berg's excellent analysis of this counterintuitive probability problem may be found here.

Educate yourself! Avoid getting Munganed out in the middle of nowhere.