Proof that neq1 is wrong:

Let H be the event that heads was flipped in this experiment instance. We're going to let Beauty experience a waking now. Let M be the event that the waking is on Monday. Let B be the information that Beauty (knowing the experiment design) has upon waking. Let h=P(H|B), and let m=P(M|B).

We wish to discover the true values of h and m. Clearly in the context of someone being asked about the expected outcome of the experiment, P(H)=1/2, but h may (or may not) differ from 1/2.

• Fact 1: P(H|M,B)=P(H)=1/2

• Fact 2: P(H|~M,B)=0 (by ~M I mean the complement of M, i.e. that it's not Monday)

Given the above two facts, we know enough to solve for h and m.

lemma 1:

• P(~H|B)=P(M,B)P(~H|M,B)+P(~M)P(~H|~M,B) ; probability axiom
• (1-h)=m(1/2)+(1-m)(1) ; by facts 1-2 and above axiom
• 1-(h)=1-(m/2) ; above simplified
• h=m/2

lemma 2:

• P(H|B)=P(M)P(H|M,B)+P(~M)(P(H|~M,B) ; probability axiom
• h=m(1/2)+(1-m)(0) ; facts 1-2 and above
• h=m/2 ; simplified
• m=2h

(oops, that turned out to be redundant; not surprising since I'm using in lemma 2 the variants p(~X)=1-P(X) from the same facts 1+2).

P(H|B) is a weighted average of the probability for heads given Monday (1/2) and given Tuesday (0). It turns out that, according to thirders, it's more likely that it's Monday (m=2h=2/3).

The thirder argument is that m=2/3 (that is, 2 out of 3 wakings on average are on Monday). The halfer argument that h=1/2 implies that m=1; that is, that Beauty is certain that it's Monday (but this is obviously stupid of her).

I was originally sympathetic to neq1's argument that B is merely "1 or more wakings occur" and that P(H|1 or more wakings occur)=P(H)=1/2, since 1 or more wakings always occur, no matter whether H or ~H. But B is better characterized as "Beauty has just been woken, not knowing whether it's the first or second waking, but knowing the experiment design".

I would like to strengthen this argument to prove that m=2/3.