Thanks for the explanation, that helped a lot. I expected you to answer 0.5 in the second scenario, and I thought your model was that total ignorance “contaminated” the model such that something + ignorance = ignorance. Now I see this is not what you meant. Instead it’s that something + ignorance = something. And then likewise something + ignorance + ignorance = something according to your model.
The problem with your model is that it clashes with my intuition (I can’t find fault with your arguments). I describe one such scenario here.
My intuition is that the probability of these two statements should not be the same:
A. “In order for us to succeed one of 12 things need to happen”
B. “In order for us to succeed all of these 12 things need to happen”
In one case we’re talking about a disjunction of 12 unknowns and in the second scenario we’re talking about a conjunction. Even if some of the “things” are not completely uncorrelated that shouldn’t affect the total estimate that much. My intuition is that saying P(A) = 1 − 0.5 ^ 12 and P(B) = 0.5 ^ 12. Worlds apart! As far as I can tell you would say that in both cases the best estimate we can make is 0.5. I introduce the assumption of independence (I don’t stipulate it) to fix this problem. Otherwise the math would lead me down a path that contradicts common sense.
Thanks for the explanation, that helped a lot. I expected you to answer 0.5 in the second scenario, and I thought your model was that total ignorance “contaminated” the model such that something + ignorance = ignorance. Now I see this is not what you meant. Instead it’s that something + ignorance = something. And then likewise something + ignorance + ignorance = something according to your model.
The problem with your model is that it clashes with my intuition (I can’t find fault with your arguments). I describe one such scenario here.
My intuition is that the probability of these two statements should not be the same:
A. “In order for us to succeed one of 12 things need to happen”
B. “In order for us to succeed all of these 12 things need to happen”
In one case we’re talking about a disjunction of 12 unknowns and in the second scenario we’re talking about a conjunction. Even if some of the “things” are not completely uncorrelated that shouldn’t affect the total estimate that much. My intuition is that saying P(A) = 1 − 0.5 ^ 12 and P(B) = 0.5 ^ 12. Worlds apart! As far as I can tell you would say that in both cases the best estimate we can make is 0.5. I introduce the assumption of independence (I don’t stipulate it) to fix this problem. Otherwise the math would lead me down a path that contradicts common sense.