Hmm, you’re right. Your math is wrong for the reason in my above comment, but the general form of the conclusion would still hold with different, weaker numbers.
The actual, more important issue relates to the circumstances of the bet:
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
If the bet would not be provided equally to all awakenings, then a thirder would update on receiving the bet.
Your math is wrong for the reason in my above comment
What exactly is wrong? Could you explicitly show my mistake?
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
The bet is proposed on every actual awakening, so indeed no update upon its receiving. However this “rational in expectation” trick doesn’t work anymore as shown by the betting argument. The bet does occur more times in high-awakening cases but you win the bet only when the maximum possible awakening happened. Until then you lose, and the closer the number of awakenings to the maximum, the higher the loss.
Hmm, you’re right. Your math is wrong for the reason in my above comment, but the general form of the conclusion would still hold with different, weaker numbers.
The actual, more important issue relates to the circumstances of the bet:
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
If the bet would not be provided equally to all awakenings, then a thirder would update on receiving the bet.
What exactly is wrong? Could you explicitly show my mistake?
The bet is proposed on every actual awakening, so indeed no update upon its receiving. However this “rational in expectation” trick doesn’t work anymore as shown by the betting argument. The bet does occur more times in high-awakening cases but you win the bet only when the maximum possible awakening happened. Until then you lose, and the closer the number of awakenings to the maximum, the higher the loss.
See my top-level comment.