In this week’s Slate Money podcast (starting at about 2:50), there’s an example of a probability puzzle that mathematically trained people tend to get wrong. In summary, the story is
You’re at a theatre watching a magician. The magician hands a pack of cards to one member of the audience and asks him to check that it is an ordinary pack, and to shuffle it. He asks another member of the audience to name a card. She says “Ace of Hearts”. The magician covers his eyes, reaches out to the pack of cards, fumbles around a bit, and pulls out a card. What’s the probability that it is the Ace of Hearts?
It’s very tempting to say 1 in 52, because the framing of the puzzle prompts you to think in terms of equal-probability sampling. Of course, as Felix Salmon points out, this is the only definitively wrong answer. The guy’s a magician. Why would he be doing this if the probability was going to be 1 in 52?
With an ordinary well-shuffled pack of cards and random selection we do know the probability: if you like the frequency interpretation of probability it’s an unknown number quite close to 1 in 52, if you like the subjective interpretation it should be a distribution of numbers quite close to 1 in 52.
With a magic trick we’d expect the probability (in the frequency sense) to be close to either zero or one, depending on the trick, but we don’t know. Under the subjective interpretation of probability then you do know what the probability is for you, but you’ve got no real reason to expect it to be similar for other people.