We are in the world where the calculator displays even, and we are 99% sure it is the world where the calculator has not made an error. This is Even World, Right Calculator. Counterfactual worlds:

• Even World, Wrong Calculator (1% of Even Worlds)
• Odd World, Right Calculator (99% of Odd Worlds)
• Odd World, Wrong Calculator (1% of Odd Worlds)

All Omega told us was that the counterfactual world we are deciding for, the calculator shows Odd. We can therefore eliminate Odd World, Wrong Calculator. Answering the question is, in essence, deciding which world we think we're looking at.

So, in the counterfactual world, we're either looking at Even World, Wrong Calculator or Odd World, Right Calculator. We have an equal prior for the world being Odd or Even - or, we think the number of Odd Worlds is equal to the number of Even Worlds. We know the ratio of Wrong Calculator worlds to Right Calculator worlds (1:99). This is, therefore, 99% evidence for Odd World. The correct decision for the counterfactual you in that world is to decide Odd World. The correct decision for you?

Ignoring Bostrom's book on how to deal with observer selection effects (did Omega go looking for a Wrong Calculator world and report it? Did Omega go looking for an Odd World to report to you? Did Omega pick at random from all possible worlds? Did Omega roll a three-sided die to determine which counterfactual world to report?), I believe the correct decision is to answer Odd World for the counterfactual world, with 99% certainty if you are allowed to specify as such.

I reason that by virtue of it being a counterfactual world, it is contingent on my not having the observation of my factual world; factual world observations are screened off by the word "counterfactual".

The other possibility (which I tentatively think is wrong) is that our 99% confidence of Even World (from our factual world) comes up against our 99% confidence of Odd World (from our counterfactual) and they cancel out, bringing you back to your prior. So you should flip a coin to decide even or odd. I think this is wrong because 1) I think you could reason from 50% in the countefactual world to 50% in the factual world, which is wrong, and 2) this setup is identical to punching in the formula, pressing the button and observing "even", then pressing the button again and observing "odd". I don't think you can treat counterfactual worlds as additional observations in this manner.

edit: It occurs to me that with Omega telling you about the counterfactual world, you are receiving a second observation. For this understanding, you would specify Even World with 99% confidence in the factual world and either Even or Odd World depending on how the coin landed for the counterfactual world.

Suppose you believe that 2+2=4, with the caveat that you are aware that there is some negligible but non-zero probability that The Dark Lords of the Matrix have tricked you into believing that.

Omega appears and tells you that in an alternate reality, you believe that 2+2=3 with the same amount of credence, and asks whether this changes your own amount of credence that 2+2=4.

The answer is the same. You ask Omega what rules he's playing by.

If he says "I'm visiting you in every reality. In each reality, I'm selecting a counterfactual where your answer is different" then you say "I have no new information, anthropic or otherwise, so I do not update."

If he says "I'm visiting you in every reality. In each reality, I'm selecting a random alternate reality where you exist and telling you what that you believes" then you say "It's equally likely that you randomly picked a reality where I am deceived and that I am in a reality where I am deceived. Therefore, I now give '2+2=4' negligibly less than a .5 chance of being true.'

I suspect that the question sounds confusing because it conflates different counterfactual worlds. Where exactly does the world presented to you by Omega diverge from the actual world, at what point does the intervention take place? If Omega only changes the calculator display, you should say "even". If it fixes an error in the calculator's inner workings, you should say "odd".

In what way, if any, is this problem importantly different from the following "less mathy" problem?

You have a sealed box containing a loose coin. You shake the box and then set it on the table. There is no a priori reason for you to think that the coin is more or less likely to have landed heads than tails. You then take a test, which includes the question: "Did the coin land heads?" Fortunately, you have a scanning device, which you can point at the box and which will tell you whether the coin landed heads or tails. Unfortunately, the opaque box presents some difficulty even to the scanning device, so the device's answer is right only 99% of the time. Furthermore, its errors are stochastic (or even involve quantum randomness), so, for any given coin-in-a-box, the device is probably correct but has a chance of making an error. You point the scanning device at the box and observe the result (it's "heads").

Then, unsurprisingly, Omega appears and presents you with the following decision. Consider the counterfactual world where the coin landed the same as it did in your world, but where the scanning device displayed "tails" instead of "heads", after you pointed it at the box. You are to determine what Omega writes on the test sheet in that counterfactual world.

I take out a pen and some paper, and work out what the answer really is. ;)

What does it even mean to write an answer on a counterfactual test sheet?

Is it correct to to interpret this as "if-counterfactual the calculator had showed odd, Omega would have shown up and (somehow knowing what choice you would have made in the "even" world) altered the test answer as you specify"?

Viewing this problem from before you use the calculator, your distribution is P(even) = P(odd) = 0.5. There are various rules Omega could be playing by:

• Omega always (for some reason uncorrelated to the parity of Q) asks you what to do iff the calculator shows even, and alters the sheet iff the calculator shows odd. Deciding "write even", your answer is only ever correct if Q is indeed even (regardless of the calculator), hence P(correct) = P(even) = 0.5. Deciding "write odd" is identical to the case without Omega (your answer is just what the calculator says) hence P(correct) = P(calculator is correct) = 0.99. The thing to do is decide to "write odd"
• Same as the above with even and odd reversed. "Leave the counterfactual paper alone" is still the correct answer.
• Some random combination of the previous two uncorrelated to the parity of Q. "Leave the counterfactual paper alone" would still be the correct answer.
• Omega knows the parity of Q, and asks you iff the calculator is correct. Deciding to say "alter the paper", the answer written is always the correct one, P(correct) = 1. "Alter the counterfactual paper" is obviously the correct answer.
• Omega asks you iff the calculator is correct, with probability p (maybe he's somehow unsure of the parity of Q, and asks you iff the calculator's result is the one he thinks is most likely). Deciding "alter the counterfactual paper", the answer written down is the correct one (again, regardless of the calculator) with probability P(correct) = p. As before, "don't alter the paper" gives you P(correct) = 0.99. Hence, answer "alter the paper" iff p > 0.99.

Finding a prior on these possibilities is left to the reader.

Why does observational knowledge work in your own possible worlds, but not in counterfactuals?

It does not work in this counterfactual. Omega could have specified the counterfactual such that the observational knowledge in the counterfactual was as usable as that in the 'real' world. (Most obviously by flat out saying it is so.)

The reason we cannot use the knowledge from this particular counterfactual is that we have no knowledge about how the counterfactual was selected. The 99% figure (as far as we know) is not at all relevant to how likely it is that we would be presented with an even or odd counterfactual result. When we intuitively reject the counterfactual result we, or at least I, are making this judgement.

Consider the following thought experiment

You have a bag with a red and a blue ball in it. You pull a ball from the bag, but don't look at it. What is the probability that it is blue?

Now imagine a counterfactual world. In this other world you drew the red ball from the bag. Now imagine a hippo eating an octopus. What is the probability that you drew the blue ball?

"Why does observational knowledge work in your own possible worlds, but not in counterfactuals?" is the key question here. Perhaps it's easier to parse like this: "Why isn't anything you can think of evidence?"

EDIT: Note that although that last question makes my answer to Vladimir's question obvious, answering the question itself requires, basically, defining what evidence is. I suppose I may as well be helpful: evidence is what you get when an event happens that lets you apply Bayes' rule to learn something new - not just any old event will do, it has to be an event that gives you different information under different circumstances.

This seems easy. Q is most likely even, so in the counterfactual the calculator is most likely in error, and we prefer Omega to write "even". What am I missing?