Sometimes there isn’t a (useful) probability
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.
Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient See all posts by Thomas Lumley »
I have many nits to pick with this post, but I’ll stick to the main one.
It’s not about “liking” the “frequency interpretation” or the “subjective interpretation” of probabilities. That’s like taking about the “biology interpretation” and the “physics interpretation” of ordinary differential equations. No, it’s just that you can use ODEs to model different kinds of things in the world. If you aren’t clear what you’re doing you’ll end up confusing everyone which ends in the situation where mathematically trained people can’t answer the simple problem in this post.
Probabilities are mathematical objects that obey certain rules. Frequencies also obey those rules, so if there are frequencies in your problem, you get some constraints about them for free. Uncertainties should obey the same rules too, so if your question involves uncertainty, you get constraints on how they can be combined and updated.
Ironically the “justification” for the common belief that “random shuffling” (which is a completely undefined concept) produces each card with frequency 1/52 is either circular reasoning (if I repeated the experiment lots of times and obtained a different frequency, you would reject the experimental evidence and claim the shuffling wasn’t random) or the result of a uniform prior over the set of possible sequences of card draws.
10 years ago
The probability in question here is the probability that he nails the conjuring trick which is mainly a function of the conjuror’s skill. Either the aim of the trick is to come up with a match, or to pretend he’s messed up as a prelude to a more impressive trick.
Framing the problem this way leads to Bayes rule and the formulation of the problem in De Fenetti type Bayesian probabilities.
10 years ago