An interesting probability paradox from Futility Closet who credits Gábor J. Székely’s Paradoxes in Probability Theory and Mathematical Statistics via's Mark Chang’s Paradoxes in Scientific Inference.
Variance in a jury's judgement seems to be better than taking one person's word for it. As Futility Closet mentions:
Chang writes, “This paradox implies it is better to have your own opinion even if it is not as good as the leader’s opinion, in general.”From Futility Closet consider:
"A, B, C, D, and E make up a five-member jury. They’ll decide the guilt of a prisoner by a simple majority vote. The probability that A gives the wrong verdict is 5%; for B, C, and D it’s 10%; for E it’s 20%. When the five jurors vote independently, the probability that they’ll bring in the wrong verdict is about 1%".For such a 5 member juries the possibilities are: mistaken=1, correct=0:
A B C D E P(A) P(B) P(C) P(D) P(E) Product
1 0 0 0 0 0.05 0.9 0.9 0.9 0.8 0.02916
0 1 0 0 0 0.95 0.1 0.9 0.9 0.8 0.06156
0 0 1 0 0 0.95 0.9 0.1 0.9 0.8 0.06156
0 0 0 1 0 0.95 0.9 0.9 0.1 0.8 0.06156
0 0 0 0 1 0.95 0.9 0.9 0.9 0.2 0.13851
1 1 0 0 0 0.05 0.1 0.9 0.9 0.8 0.00324
1 0 1 0 0 0.05 0.9 0.1 0.9 0.8 0.00324
1 0 0 1 0 0.05 0.9 0.9 0.1 0.8 0.00324
1 0 0 0 1 0.05 0.9 0.9 0.9 0.2 0.00729
0 1 1 0 0 0.95 0.1 0.1 0.9 0.8 0.00684
0 1 0 1 0 0.95 0.1 0.9 0.1 0.8 0.00684
0 1 0 0 1 0.95 0.1 0.9 0.9 0.2 0.01539
0 0 1 1 0 0.95 0.9 0.1 0.1 0.8 0.00684
0 0 1 0 1 0.95 0.9 0.1 0.9 0.2 0.01539
0 0 0 1 1 0.95 0.9 0.9 0.1 0.2 0.01539
0 0 1 1 1 0.95 0.9 0.1 0.1 0.2 0.00171
0 1 0 1 1 0.95 0.1 0.9 0.1 0.2 0.00171
0 1 1 0 1 0.95 0.1 0.1 0.9 0.2 0.00171
0 1 1 1 0 0.95 0.1 0.1 0.1 0.8 0.00076
1 0 0 1 1 0.05 0.9 0.9 0.1 0.2 0.00081
1 0 1 0 1 0.05 0.9 0.1 0.9 0.2 0.00081
1 0 1 1 0 0.05 0.9 0.1 0.1 0.8 0.00036
1 1 0 0 1 0.05 0.1 0.9 0.9 0.2 0.00081
1 1 0 1 0 0.05 0.1 0.9 0.1 0.8 0.00036
1 1 1 0 0 0.05 0.1 0.1 0.9 0.8 0.00036
0 1 1 1 1 0.95 0.1 0.1 0.1 0.2 0.00019
1 0 1 1 1 0.05 0.9 0.1 0.1 0.2 0.00009
1 1 0 1 1 0.05 0.1 0.9 0.1 0.2 0.00009
1 1 1 0 1 0.05 0.1 0.1 0.9 0.2 0.00009
1 1 1 1 0 0.05 0.1 0.1 0.1 0.8 0.00004
1 1 1 1 1 0.05 0.1 0.1 0.1 0.2 0.00001
All those possibilities in red are mistaken coalitions with probability totaling: 0.00991.
[This is slightly smaller than the result originally posted which over-estimated this value as a comment suggested.]
From Futility Closet:
"But if E (whose judgment is poorest) abandons his autonomy and echoes the vote of A (whose judgment is best), the chance of an error rises to 1.5%".In this situation juror E always agrees with juror A, so if A is included in a mistaken coalition it only needs two more jurors to form a simple majority. Of course A might not be included, then a mistaken coalition needs jurors B, C, and D. The possibilities and their probabilities are shown below:
A B C D P(A) P(B) P(C) P(D) Product
1 0 0 0 0.05 0.9 0.9 0.9 0.03645
0 1 0 0 0.95 0.1 0.9 0.9 0.07695
0 0 1 0 0.95 0.9 0.1 0.9 0.07695
0 0 0 1 0.95 0.9 0.9 0.1 0.07695
1 1 0 0 0.05 0.1 0.9 0.9 0.00405
1 0 1 0 0.05 0.9 0.1 0.9 0.00405
1 0 0 1 0.05 0.9 0.9 0.1 0.00405
0 1 1 0 0.95 0.1 0.1 0.9 0.00855
0 1 0 1 0.95 0.1 0.9 0.1 0.00855
0 0 1 1 0.95 0.9 0.1 0.1 0.00855
0 1 1 1 0.95 0.1 0.1 0.1 0.00095
1 0 1 1 0.05 0.9 0.1 0.1 0.00045
1 1 0 1 0.05 0.1 0.9 0.1 0.00045
1 1 1 0 0.05 0.1 0.1 0.9 0.00045
1 1 1 1 0.05 0.1 0.1 0.1 0.00005
All those possibilities in red are mistaken coalitions with probability totaling: 0.0145.
[This is slightly smaller than the result originally posted as a comment suggested.]
Again from Futility Closet:
"Even more surprisingly, if B, C, D, and E all follow A, then the chance of a bad verdict rises to 5%, five times worse than if they vote independently, even though A is nominally the best leader".Variance is good!
4 comments:
Isn't there a slight overcount there? If ABCD all agree, you've counted their mistake twice (three times if ABCDE agree).
You are right Kevin. Thanks for catching my mistake. I've corrected it here.
I did a generalization of the problems with only three judges here:
http://www.cphpvb.net/probability/9251-when-to-follow-authority/
You are missing the "0 0 0 0 0" scenario in the article.
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