jessicata comments on Sleeping Beauty Resolved (?) Pt. 2: Identity and Betting

jessicata 3 Jun 2018 3:15 UTC
3 points
I think your equation $P r (H | M) = \sum_{i} P r (X_{2} (i) | M) \cdot p$ is incorrect [EDIT: actually correct but wouldn’t be for computing probability of tails, see bottom]. I assume you intend $p$ to mean $P r (H | X_{2} (0), M)$ (which equals $P r (H | X_{2} (1), M)$ ). So the equation says $P r (H | M) = \sum_{i} P r (X_{2} (i) | M) \cdot P r (H | X_{2} (i), M)$ . But, as you point out, $X_{2} (0)$ and $X_{2} (1)$ are not mutually exclusive. So this equation doesn’t follow from probability theory.
In general if you have events $A, B, C$ where $B$ and $C$ are not mutually exclusive, then it is not necessarily the case that $P r (A) = P r (B) \cdot P r (A | B) + P r (C) \cdot P r (A | C)$ . For example, say that $A, B, C$ are all always true. Then the equation says $1 = 1 \cdot 1 + 1 \cdot 1$ .
EDIT: the equation is actually true. This is because because $X_{2} (0)$ and $X_{2} (1)$ are mutually exclusive given H. But the equation would be false if $H$ were replaced with $T$ in the equation, since $X_{2} (0)$ and $X_{2} (1)$ are not mutually exclusive given T.
- ksvanhorn 9 Jun 2018 16:52 UTC
  4 points
  Parent
  You’re right, my argument wasn’t quite right. Thanks for looking into this and fixing it.