In the Mad Bayesian’s Lair

September 14, 2010

Here’s a thought experiment on Bayes’s Rule that really captured my imagination:

Someone hits you over the head and knocks you out. You wake up to find yourself in a brightly-lit, all-white room, like in a futuristic sci-fi movie. In the room you see two doors with a single letter, “A” and “B” in large black type on each one. The only things in the room itself are a solidly-built, polished wooden box and a note. The note reads:

Hello and Welcome to my room. As you can see, there’s not much in it, but the doors are another story. Behind one of the doors you will find your heart’s deepest desire. Behind the other, your greatest fear. Obviously, the correct choice for you is the one that leads to your deepest desire. But be warned: as soon as you open one of the doors, you will be forced out and the other option will be closed to you forever. So choose wisely.

I’ll tell you, first of all, that I have prepared 100 rooms exactly like the one you are in. In exactly half of these rooms, door “A” leads to your deepest desire, while “B” leads you two your doom. In the other half, the doors are switched.

I’m going to tell you something else. Within the box you will find a single playing card, the King of Hearts or the Queen of Spades. Now pay attention: in 80% of the rooms where “A” is the door you want, I have placed the King in the box. But I have also done that in 40% of the rooms where “B” is the correct choice. Contrarily, in 20% of the rooms where “A” is correct, I have placed a Queen in the box. I have done the same in 60% of the rooms where “B” is correct. That’s all I’m going to tell you.

Sincerely yours,

A crazy person

Upon closer examination, you notice that the letter is signed and notarized by your mother, which gives you confidence that everything written in it is 100% accurate. So all that’s left to do is open the box and make your choice.

Questions: If you find a King, what are the odds that door “A” is the correct choice? If you find a Queen, what are the odds that it’s door “A”?


2 Responses to “In the Mad Bayesian’s Lair”

  1. Zac Gochenour Says:

    P(A|King) = P(King|A)*P(A)/P(King) = .8 * .5 / (.8 * .5 + .4 * .5) = 2/3

    P(A|Queen) = .2 * .5 / (.2 * .5 + .6 * .5) = 1/4

    P(B|King) = .4 * .5 / (.8 * .5 + .4 * .5) = 1/3

    p(B|Queen) = .6 * .5 / (.2 * .5 + .6 * .5) = 3/4

    It is interesting. The results aren’t really intuitive, but it confirmed my initial reaction that I’d rather see the Queen.

  2. Thanks for answering! I was boycotting this blog until someone did.

    The thing that annoys me about this puzzle is that in the end, you just do what you would have done if you didn’t know anything about Bayesian probability, you just do it with more knowledge. So it’s not very Monty Hall-esque.

    The reason I like the structure is that it makes concrete the idea that there are many universes, and in some of them you observe certain signals more often, in others you observe other signals. And the real problem is figuring out what universe you’re in.

    I guess I could have a doctor put you in a room and screen you for a rare disease…

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