thomblake comments on Conditioning on Observers - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (118)
Another Rival Intution Pump
Suppose that for exactly one day every week you sleep during the day and wake up in the evening, and for every other day you sleep at night and wake up in the morning.
Suppose that for a minute after waking, you can reason logically but cannot remember what day it is (and have no way of telling the time).
Then during that minute, surely your subjective probability of it being morning is 6/7.
OK now let's change things up a little:
At the beginning of every week, a coin is flipped. If heads then rather than having 6 days diurnal and 1 day nocturnal, you just have 1 day nocturnal and six days in hibernation. If tails then you have 6 days diurnal and 1 day in hibernation.
Then surely the addition of the coin flip and the hibernation doesn't change the fact that (for any given awakening) you have a 6/7 probability of waking in the morning.
Well, consider:
Scenario 1: You have a bag containing one red ball and an arbitrarily large number of green balls. You reach in and pull out one ball at random. What is the probability that the ball is red?
Scenario 2: You have a bag containing one red ball and another bag containing an arbitrarily large number of green balls. A fair coin is flipped; if heads, you are handed the bag with the red ball, and if tails you are handed the bag with the green balls (you can't tell the difference between the bags). You reach in and pull out one ball at random. What is the probability that the ball is red?
In scenario 1, P(red) is vanishingly small. In scenario 2, P(red) is 1/2.
The disanalogy is that you actually pull out all of the green balls, not just one.
Indeed - introducing amnesia and pulling out each of the green balls in turn might muddy this one up as well.