But with the conditions given, if someone says "yes", there's an 80% chance they're guilty.
Don't confuse P(A|B) with P(B|A)! If the real proportion of guilty people is p, then 5p/6 of people will truthfully answer yes (that includes the ones who rolled a six and were guilty) and (1-p)/6 of people will falsely answer yes. So out of all the yes answers, the proportion of guilty people is 5p / (4p + 1), which varies monotonically with p, can itself be anything from 0 to 1, and is only equal to 80% if p=4/9.
I can see your reasoning for the 80% figure: if someone says yes, we know they didn't roll a 1, therefore there are five remaining equiprobable things they might have rolled and four of those five mean they're guilty as hell. The flaw is in the word "equiprobable": because rolling a 6 changes your probability of saying yes, it follows that if all you know about someone is that they said yes, it's no longer equiprobable that they rolled 2, 3, 4, 5 or 6.
It's still high-ish, but less than half (if I did the revised calcualtion right, and the percentage of people who actually said yes was accurately deduced and reported in the article)
I would add that we can get a point estimate of p from sufficient samples using:
Prob (Yes) = (1+4p)/6 and Prob (No) = (5-4p)/6
Hence if we know the number of nos and yeses we can estimate the probability of guilt given a yes and a no answer and put confidence intervals on it should we so choose (if we know the sample size.
I think I was subconsciously assuming you were doing this with something where everyone was guilty, whereas that's not true here: it's endemnic, but that still means only 20% of people.
Indeed, if absolutely everyone were guilty and the aim of this exercise was purely to try to trick them into confessing something that could be used in a court of law, your reasoning would be perfectly all right :-)
I seem to recall teh version I heard was something like "do you masterbate" or "have you ever been influenced by a source you failed to cite when writing an essay at school" which applies to almost everyone, but isn't illegal.
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andrewducker) wrote2011-08-01 12:40 pm
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Don't confuse P(A|B) with P(B|A)! If the real proportion of guilty people is p, then 5p/6 of people will truthfully answer yes (that includes the ones who rolled a six and were guilty) and (1-p)/6 of people will falsely answer yes. So out of all the yes answers, the proportion of guilty people is 5p / (4p + 1), which varies monotonically with p, can itself be anything from 0 to 1, and is only equal to 80% if p=4/9.
I can see your reasoning for the 80% figure: if someone says yes, we know they didn't roll a 1, therefore there are five remaining equiprobable things they might have rolled and four of those five mean they're guilty as hell. The flaw is in the word "equiprobable": because rolling a 6 changes your probability of saying yes, it follows that if all you know about someone is that they said yes, it's no longer equiprobable that they rolled 2, 3, 4, 5 or 6.
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It's still high-ish, but less than half (if I did the revised calcualtion right, and the percentage of people who actually said yes was accurately deduced and reported in the article)
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I would add that we can get a point estimate of p from sufficient samples using:
Prob (Yes) = (1+4p)/6
and
Prob (No) = (5-4p)/6
Hence if we know the number of nos and yeses we can estimate the probability of guilt given a yes and a no answer and put confidence intervals on it should we so choose (if we know the sample size.
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