4 ms·

1/36 is the probability that the detector is lying, and so when it says "yes" that means it's a 1/36 probability that the sun hasn't gone nova, not the probabil

1/36 is the probability that the detector is lying, and so when it says "yes" that means it's a 1/36 probability that the sun hasn't gone nova, not the probability that it could. $50 is only really any use if the sun hasn't gone nova (cause otherwise we're all dead), so therefore even if the odds are bad it's still a good bet. EDIT: Also, a penny just dropped after reading another of the answers. The odds against the sun having gone nova are incredibly mind-bogglingly high such that the the 1/36 probability is negligible. So even if the probability that the machine is lying is small at 1/36 it's still far far far more likely than the sun having gone nova - which is the "prior" that the Bayesian is taking into account. Another rub might also be that the frequentist's significance (P-value) is not sufficient to take into account the astronomical scale of the Nova probability but IANAS so that's just speculation.

raverbashing 11y ago
Yes, what I mean is that, if you got a "yes" from the detector, it either means that: - The sun hasn't gone nova and the detector is lying (with 1/36 chance) - The sun has gone nova and the detector is not lying (negligible) Now, given the result, one can calculate the probability of which of the cases And the betting aspect of it makes sense as well
- bonoboTP 11y ago
  The "1/36 chance" and the "negligible" aren't the corresponding/analogous probabilities. 1/36 is P(detector "yes" | not nova). The "negligible" is P(nova). What we want to calculate is P(nova | detector "yes"). Using Bayes' rule it's P(nova | detector "yes") = P(detector "yes" | nova) P(nova) / (P(detector "yes" | nova) P(nova) + P(detector "yes" | not nova) P(not nova)) = 35/36 * P_nova / (35/36 * P_nova + 1/36 * (1-P_nova)) P(nova | detector "yes") > 0.5 if P_nova > 1/36. This is also quite intuitive. Unless going nova is more likely than malfunctioning, you should bet on malfunctioning. This is very much like David Hume's quote: "no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish." (test: would you say the same amount of "holy crap! WTF!!" if it turned out that the testimony was false, compared to if it turned out that the miraculous thing actually happened). So: no detector "yes" is sufficient to establish the sun going nova, unless the detector saying "yes" be such, that it lying (1/36) would be more miraculous, than the sun going nova. It's fascinating how precisely this can be formulated and how clear it is with probability theory.

raverbashing 11y ago

Yes, what I mean is that, if you got a "yes" from the detector, it either means that: - The sun hasn't gone nova and the detector is lying (with 1/36 chance) - The sun has gone nova and the detector is not lying (negligible) Now, given the result, one can calculate the probability of which of the cases And the betting aspect of it makes sense as well

bonoboTP 11y ago
The "1/36 chance" and the "negligible" aren't the corresponding/analogous probabilities. 1/36 is P(detector "yes" | not nova). The "negligible" is P(nova). What we want to calculate is P(nova | detector "yes"). Using Bayes' rule it's P(nova | detector "yes") = P(detector "yes" | nova) P(nova) / (P(detector "yes" | nova) P(nova) + P(detector "yes" | not nova) P(not nova)) = 35/36 * P_nova / (35/36 * P_nova + 1/36 * (1-P_nova)) P(nova | detector "yes") > 0.5 if P_nova > 1/36. This is also quite intuitive. Unless going nova is more likely than malfunctioning, you should bet on malfunctioning. This is very much like David Hume's quote: "no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish." (test: would you say the same amount of "holy crap! WTF!!" if it turned out that the testimony was false, compared to if it turned out that the miraculous thing actually happened). So: no detector "yes" is sufficient to establish the sun going nova, unless the detector saying "yes" be such, that it lying (1/36) would be more miraculous, than the sun going nova. It's fascinating how precisely this can be formulated and how clear it is with probability theory.