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Zvi comments on Bead Jar Guesses - Less Wrong
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You should not guess that the first bead has a 50% chance of being red, because if you do, you can have this conversation:
Omega: What is the probability of the first bead being red as opposed to non-red?
You: Fifty-fifty.
Omega: So you would consider it more than fair if I offered you three dollars if the bead is red, and you paid me a dollar if it was non-red?
You: Sure, I'll take that bet.
Omega: What is the probability of the first bead being blue as opposed to non-blue?
You: Fifty-fifty.
Omega: So you would consider it more than fair if I offered you three dollars if the bead is blue, and you paid me a dollar if it was non-blue?
You: Sure, I'll take that bet.
(...and so on for ten more colors.)
Omega pulls out a red bead. He owes you three dollars, but you owe him eleven dollars. He wins.
You could have that conversation, but you don't have to. The argument for assigning 50% to red is that it's the only question Omega has asked you. There are several ways out of that. The first one is that the moment he offers you a 25% bet I would update to presume that 3:1 is not a positive e.v. bet, with a new number of perhaps 12.5% with a range of 0% to 25% with symmetric distribution. Similarly, if he offered me three to one that it wasn't red, I would presume that it probably will be. On a similar note, when he asks about blue (even without any bets involved) I can't see answering higher than 33.3%.
Contrast this with Alicorn watching this incident and offering me 3:1 after Omega asks my probability for red and I say 50%. I still have to update for Alicorn's opinion, but I might or might not accept that bet.