The Tao of Gaming

Boardgames and lesser pursuits

An Interesting question about probability…

You are in the audience at a small, intimate theatre, watching a magic show. The magician hands a pack of cards to a random member of the audience, asks him to check that it’s an ordinary pack, and would he please give it a shuffle. The magician turns to another member of the audience and asks her to name a card at random. “Ace of Hearts,” she says. The magician covers his eyes, reaches out to the pack of cards, and after some fumbling around he pulls out a card. The question to you is what is the probability of the card being the Ace of Hearts?

Answer in the comments and then read the whole article.

(I could argue that this is related to game strategy, and it may be, but I really just like the arguments this causes.)

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Written by taogaming

December 17, 2008 at 9:48 pm

Posted in Open Thread

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17 Responses

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  1. Erm. I’d guess the magician is going to pull the right card nearly all of the time if he wants, but it seems like half the time the trick involves pulling the “wrong card” once or twice as a known gag. So I’ll throw out a 75%. Surely that “half the time” is overestimating.

    GreedyAlgorithm

    December 18, 2008 at 4:45 am

  2. For a randomly selected card, we should expected a 1 in 52 chance that the named card is selected, assuming it really is a random draw from a 52 card deck.

    Dan

    December 18, 2008 at 8:36 am

  3. Rob @1: I figure between screwing it up accidentally, and “screwing it up” deliberately as a setup for something else, 90%’s probably reasonable. At least, it’s the number I pulled out of a hat.

    Tucker

    December 18, 2008 at 8:50 am

  4. Now I’ve read it, and I am not impressed. If Wilmott is trying to make a point, then has missed the obvious, and only added confusion and misinformation. Mathematics handles this problem just fine, if you take into account all the information you have.

    The answer to the stated question is 1 in 52. The magician does depend on this probability, not as the result, but as the expectation of the audience which makes the trick interesting.

    The real question is an exercise in conditional probability, and had Wilmott asked a statistician he would have gotten a better answer. I find it difficult to believe that any mathematicians worth their salt missed this too.

    Here is a better take on this problem:
    http://en.wikipedia.org/wiki/Monty_Hall_problem
    Conditional probability can lead to tricky problems, but Wilmott is just wrong to suggest that the answer lies “outside the box of mathematical theories”.

    Dan

    December 18, 2008 at 9:03 am

  5. The question is unanswerable due to lack of details. 1/52, 90%, 100% are all guesses and we have no way of knowing if any are more correct than the others.

    MrHen

    December 18, 2008 at 9:10 am

  6. I think the article has a point, but I’m not nearly as alarmed by it as the author is. Part of it is, you give a man a hammer and everything looks like a nail to him. Ask a mathematician about a problem that seems to involve probabilities and it isn’t surprising that he tries to solve it using that discipline.

    It’s easy to see that, given the conditions of the question, that the answer might be other than 1/52. The thing is, once you rule that out, the problem is no longer of any interest. The magician will probably “cheat” somehow, but in ways that can’t be estimated. So the question has no calculable answer. Really, the only other answer that makes any sense is 100%, based on the reasoning of “it’s a magician’s trick, he’s in control, and it isn’t likely he’d be performing it if he couldn’t produce the desired result”. That would be my answer, but my follow-up response would be “why are you wasting my time with this pointless question?”.

    I guess if I mentioned this reasoning to someone and they still insisted on an answer of 1/52, I might be concerned. Comparing something elementary (like a magician’s tendency toward trickery) with something mind-numbingly complex (like the effect of human emotions on the market) seems very naive, but there is a small amount of truth there. Certainly, one of the things that’s led to this financial mess is lack of consideration of the human element (although there was also plenty of analytic mistakes made as well).

    By the way, if our “1/52” responder is basing his or her answer on the “impossibility” of the trick, I can think of several ways the magician could pull this off. Maybe the deck is stripped (which wouldn’t be detected on a cursory analysis) and he can use this to locate certain cards. Maybe the deck is in a random looking, but predetermined order and a single riffle shuffle can be undone (via stripping, for example) to get the deck back to its original order, from where the magician can find any card he wants. Maybe he’s counting on certain cards being called for (I assume the probability of the “Ace of Spades” being called out is far higher than 1/52). None of this helps with estimating a probability, but it shows that 1/52 is probably the wrong answer.

    Larry Levy

    December 18, 2008 at 10:59 am

  7. The article makes a good point – if you use simple analysis you can get a number, but all it’s done is given you a false sense of confidence. In fact, the magician is banking on this sense of confidence to make the trick impressive. If you don’t take into consideration all the variables, then your number is worse than a blind stab – and if you are using this guess for risk management, then you’re making decisions on faulty data under the illusion that you’ve got rock-solid statistical information.

    I don’t agree that coming up with a decently accurate answer is impossible, but shows that often more data is needed to refine the statistical model.

    SeanP

    December 18, 2008 at 11:12 am

  8. Not to be too much of a geek, but the 1:52 is obviously wrong. It’s got to be 1:54, right? There are two Jokers?

    The question itself is a bit of a magic trick, in that it’s all about misdirection. With a short question stated in the abstract, we don’t really know what he’s getting at. Is it a question really about the magician, or the pack of cards? Hard to tell. Most of us know the answer is either 1:52 or a lot higher, depends on what he’s looking for, and the questioner can say you’re wrong however you answer, so we sort of guess what he wants.

    The real problem with the linked article, and where he loses me, is when he says there is no right answer. Clearly what a real risk manager would do is go out and observe a bunch of magicians performing this trick, figure out how often they can get it right, and put that number into their risk management formulae. It’s a simplistic explanation of a complicated problem. Clearly risk managers screw up and get the numbers wrong. But this article does little to illuminate why that is. They missed that there was a magician in the room? Does he seriously want risk managers to use their instincts rather than math? Arguably we’re in this spot because risk managers decided that historically housing prices don’t go down and that their instincts told them that the rules had changed, when a reasonably analytical look at the risks would have told them things were unsustainable.

    Chris Farrell

    December 18, 2008 at 1:33 pm

  9. Ok ok, qualify my answer as a standard and “ordinary” (not tricked out) 52 card deck. I was answering the question about the expected probability. How the magician does the trick was not the question (but would be relevant information *if* we knew it).

    Sean and Chris are correct, the estimate must include all the information (be conditional on, as in Bayes theorem) to be accurate. Without all the correct information, we are essentially left trying to answer the wrong question.

    So how is this relevant to game strategy? I would suggest that the player with more information (here the magician) is more likely to win. A player who is familiar the rules and every aspect of play will do better than another who doesn’t fully understand how the game works. In analogy to the “Ace of Hearts” example, the magician is making up rules to the game as he goes along, and the audience is trying to figure out how to play.
    In context of typical games we might play, an experienced player has more information than a beginner, and so is more likely to win (all other things being equal).

    Dan

    December 18, 2008 at 2:52 pm

  10. Forgot to mention – There are other ways for information to be unequal between players besides just experience and knowledge of the game. Games that implement “Fog of War” or hidden movement are an examples of this (as is just paying closer attention). Again, the advantage goes to the player with the better information.

    Dan

    December 18, 2008 at 3:00 pm

  11. A good way for risk mitigation would be to do further research before blindly jumping in with statistics — observe the magician at a number of shows. Is hit routine consistent? Does he get the card right the majority of the time? How often does he change up his routine? Is the same card consistently picked by the “random” audience member? As more of these questions can be answered, and more behavior observed, a better prediction can be honed in on.

    SeanP

    December 18, 2008 at 3:52 pm

  12. Warning – this has nothing to do with the article in question. Except that is has to do with probability, which made me think to tell it.

    My older son had a band concert last night. One of the other bands played a song titled A+ . After an introduction, the piece consists of a nice little march. For the second half of the piece, the march is repeated – but this time, instead of playing everything perfectly, the band performs 97%, 98%, or 99% correctly. In this instance, the band chose 99%; the director asked what we expected to hear. I noted to my wife that she (never having played an instrument) would clearly notice the difference, even though it meant that each performer would make only one mistake through the piece. She thought I would notice, but she wouldn’t. Very enjoyable piece, I must say…

    Joe Huber

    December 18, 2008 at 5:28 pm

  13. Joe,

    You told us your predictions, but not the outcome!

    SeanP

    December 18, 2008 at 5:49 pm

  14. Yeah, what happened Joe? I can’t believe I wouldn’t have noticed the difference of even one error each, although I suspect I’d only catch about 10-20% of the errors.

    Lou

    December 18, 2008 at 11:24 pm

  15. Sorry ’bout that – started thinking about how to describe the outcome, got pulled off onto other matters, got back and didn’t recall that I hadn’t really finished the story.

    The result – with just a 1% error rate – was noticably the same piece. But the differences were clear and obvious – during the main portion of the march, there was one off note per measure or so; some were subtle enough to escape mostly unnoticed, but anything that was held for a couple of beats or longer was _very_ noticable. (One sax player overplayed a bit, playing not only the wrong note, but playing it fortissimo.) But then there was a nice, carefully timed, picasso section in which every error was not only clear, but obvious. The bass drum player dropping his stick was a nice touch.

    I shudder at the thought of 97% accuracy…

    Joe Huber

    December 19, 2008 at 10:57 am

  16. Sorry ’bout that – started thinking about how to describe the outcome, got pulled off onto other matters, got back and didn’t recall that I hadn’t really finished the story.

    The result – with just a 1% error rate – was noticably the same piece. But the differences were clear and obvious – during the main portion of the march, there was one off note per measure or so; some were subtle enough to escape mostly unnoticed, but anything that was held for a couple of beats or longer was _very_ noticable. (One sax player overplayed a bit, playing not only the wrong note, but playing it fortissimo.) But then there was a nice, carefully timed, picasso section in which every error was not only clear, but obvious. The bass drum player dropping his stick was a nice touch.

    I shudder at the thought of 97% accuracy…

    I haven’t heard the piece, but I have a feeling it wouldn’t be much worse at 98 or 97%. The initial set of errors are immediately obvious and throw the piece off, adding extras after that has less effect.

    frunk

    December 19, 2008 at 10:04 pm

  17. Not to be too much of a geek, but the 1:52 is obviously wrong. It’s got to be 1:54, right? There are two Jokers?

    As someone who’s done a good deal of reading on magic (and dabbled in it in his youth), just about every effect I can think of uses a deck without Jokers. There’s no particular reason why this has to be the case, it just is. Maybe because including Jokers can only spoil the impression of the trick; if the spectator selects a card and it turns out to be a Joker, he’s liable to suspect trickery, even if his selection is truly random. Of course, if the trick starts with a factory sealed deck, that’s different, but that isn’t typically done (unless the magician is taking advantage of the preordering of the deck).

    Larry Levy

    December 20, 2008 at 11:26 am


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