for decision-theoretic purposes you want the probability to be 1⁄3 as soon as the AI wakes up on Monday/Tuesday.
That is based on a flawed decision analysis that fails to account for the fact that Beauty will make the same choice, with the same outcome, on both Monday and Tuesday (it treats the outcomes on those two days as independent).
So you want to use FDT, not CDT. But if the additional data of which direction the fly is going isn’t used in the decision-theoretic computation, then Beauty will make the same choice on both days regardless of whether she has seen the fly’s direction or not. So according to this analysis the probability still needs to be 1⁄2 after she has seen the fly.
That is based on a flawed decision analysis that fails to account for the fact that Beauty will make the same choice, with the same outcome, on both Monday and Tuesday (it treats the outcomes on those two days as independent).
So you want to use FDT, not CDT. But if the additional data of which direction the fly is going isn’t used in the decision-theoretic computation, then Beauty will make the same choice on both days regardless of whether she has seen the fly’s direction or not. So according to this analysis the probability still needs to be 1⁄2 after she has seen the fly.