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 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.