I think I agree completely with your first paragraph. There’s no need to introduce indexical complexity, and classical probability copes with this just fine.
In all cases, the probability is 50% (or, once known, it’s 1 or 0). The casual discussion of betting methodology conflates probability and payouts. If there’s a 50% that you’ll lose twice and a 50% chance that you’ll win once, you should require a 2:1 payout on the bet, which leads people to say 33% probability when they combine the two. If you’ll only lose once EVEN WHEN ASKED TWICE, then 1:1 odds are fine and 50% is clear.
I don’t know what you mean by “in all cases, the probability is 50%”. What situation are you referring to? I’m arguing that Beauty’s probability of Heads should be 1⁄3 when woken on Monday or Tuesday, and I believe this is quite consistent with obtaining good results in any betting scenario, when the decision is made properly based on this probability. If you think Beauty’s probability for Heads should be 1⁄2, you need to do more than just assert this.
I think I agree completely with your first paragraph. There’s no need to introduce indexical complexity, and classical probability copes with this just fine.
In all cases, the probability is 50% (or, once known, it’s 1 or 0). The casual discussion of betting methodology conflates probability and payouts. If there’s a 50% that you’ll lose twice and a 50% chance that you’ll win once, you should require a 2:1 payout on the bet, which leads people to say 33% probability when they combine the two. If you’ll only lose once EVEN WHEN ASKED TWICE, then 1:1 odds are fine and 50% is clear.
I don’t know what you mean by “in all cases, the probability is 50%”. What situation are you referring to? I’m arguing that Beauty’s probability of Heads should be 1⁄3 when woken on Monday or Tuesday, and I believe this is quite consistent with obtaining good results in any betting scenario, when the decision is made properly based on this probability. If you think Beauty’s probability for Heads should be 1⁄2, you need to do more than just assert this.