… but it isn’t, because the degree of surprise doesn’t just depend on the raw probability, but also only the number of other possible outcomes under consideration. That Omega uses the term “lilac” may reasonably be taken as evidence that the space of color outcomes should be treated as finely divided.
ETA: I guess the mistake is in comparing feelings of surprise across outcomes with the same probability embedded in event spaces with different cardinalities.
If Omega asked me the probability of the next bead being lilac, I would be surprised to if the next bead actually was lilac, in a way I would not be surprised to find the bead is turquoise, an event to which I assign equal probability, but was not specifically considering prior to the draw, as any higher probability set of events which excludes drawing a turquoise bead would seem artificial. If the first two beads are the colors Omega asks me about, my leading theory would be that Omega will draw out a bead of which ever color he just brought up. (The first draw would cause me to consider this with roughly equal probability as maximum entropy.)
“doesn’t just depend on the raw probability”—Correct.
It also depends strongly on how reliable you think your estimate of the probability is.
That is, your confidence interval.
… but it isn’t, because the degree of surprise doesn’t just depend on the raw probability, but also only the number of other possible outcomes under consideration. That Omega uses the term “lilac” may reasonably be taken as evidence that the space of color outcomes should be treated as finely divided.
ETA: I guess the mistake is in comparing feelings of surprise across outcomes with the same probability embedded in event spaces with different cardinalities.
If Omega asked me the probability of the next bead being lilac, I would be surprised to if the next bead actually was lilac, in a way I would not be surprised to find the bead is turquoise, an event to which I assign equal probability, but was not specifically considering prior to the draw, as any higher probability set of events which excludes drawing a turquoise bead would seem artificial. If the first two beads are the colors Omega asks me about, my leading theory would be that Omega will draw out a bead of which ever color he just brought up. (The first draw would cause me to consider this with roughly equal probability as maximum entropy.)
“doesn’t just depend on the raw probability”—Correct. It also depends strongly on how reliable you think your estimate of the probability is. That is, your confidence interval.
Well, maybe it isn’t, but it should.