The hundred-room problem

APMason

This thought-experiment has been on my mind for a couple of days, and no doubt it's a special case of a more general problem identified somewhere by some philosopher that I haven't heard of yet. It goes like this:

You are blindfolded, and then scanned, and ninety-nine atom-for-atom copies of you are made, each blindfolded, meaning a hundred in all. To each one is explained (and for the sake of the thought experiment, you can take this explanation as true (p is approx. 1)) that earlier, a fair-coin was flipped. If it came down heads, ninety-nine out of a hundred small rooms were painted red, and the remaining one was painted blue. If it came down tails, ninety-nine out of a hundred small rooms were painted blue, and the remaining one was painted red. Now, put yourself in the shoes of just one of these copies. When asked what the probability is that the coin came down tails, you of course answer “.5”. It is now explained to you that each of the hundred copies is to be inserted into one of the hundred rooms, and will then be allowed to remove their blindfolds. You feel yourself being moved, and then hear a voice telling you you can take your blindfold off. The room you are in is blue. The voice then asks you for your revised probability estimate that the coin came down tails.

It seems at first (or maybe at second, depending on how your mind works) that the answer ought to be .99 – ninety-nine out of the hundred copies will, if they follow the rule “if red, then heads, if blue then tails”, get the answer right.

However, it also seems like the answer ought to be .5, because you have no new information to update on. You already knew that at least one copy of you would, at this time, remove their blindfold and find themselves in a blue room. What have you discovered that should allow you to revise your probability of .5 to .99?

And the answer, of course, cannot be both .5 and .99. Something has to give.

Is there something basically quite obvious that I'm missing that will resolve this problem, or is it really the mean sonofabitch it appears to be? As it goes, I'm inclined to say the probability is .5 – I'm just not quite sure why. Thoughts?

And the answer, of course, cannot be both .5 and .99. Something has to give.

I'm going to try and slowly introduce indexicality into the setup, to argue that it's really there all along and you ought to answer 0.99.

Do you have to be copied? The multiple coexisting copies are a bit of a distraction. Let's say the setup is that you're told you are going to be put in the first room and subsequently given a memory-erasing drug; then the second room and so on.

Now one more adjustment: the memory-erasing drug is imperfect; it merely makes your memory of each trial very fuzzy and uncertain. You can't remember what you saw and deduced each time, but you remember that it happened. In particular, you know that you're in the 65th room right now, although you have no idea what you saw in the previous 64.

Finally, instead of moving you from the first room to the 100th orderly, let's say that a random room is chosen for you every time, and this is repeated a million times.

In this case, the answer 0.99 seems inevitable. The choice of the room for you was randomly independent in this particular trial, and brings you fresh legitimate evidence to bayes upon. Note that you can say that you would, with very high probability, see blue in one of the trials regardless of what the coin showed. This doesn't faze you; if the coin were heads, it's very unlikely that this particular 65th trial would be one to turn blue on you; and your answer reflects this unlikelyhood.

Now go back to orderly traversing the 100 rooms. This time you can say with certainty that you'd see blue in one of the trials no matter what. But this certainty is only very very slightly more than the very high probability of the previous paragraph; going from 0.99 to 0.5 on the strength of that change is hardly warranted. As before, if the coin were heads, it's very unlikely that this particular 65th trial would be the one with the blue room; and your answer should reflect this unlikelyhood.

Now go back yet again: they improved the drug and you no longer remember how many trials you had before this one. Should this matter a lot to your response? When you knew this was the 65th trial, you knew nothing about your experience in the previous 64. And it was guaranteed that you'd eventually reach the 65th. It seems that the number of the trial isn't giving you any real information, and your answer ought to remain unchanged without access to it.