## Monday, August 19, 2013

### Variance Rules

[Earlier this post had errors. Thanks Kevin. I was thinking sequentially instead of group-wise.  For correct reference, my mistake is corrected here. The overall conclusions have not changed.]

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!

Kevin said...

Isn't there a slight overcount there? If ABCD all agree, you've counted their mistake twice (three times if ABCDE agree).

Robert W. Jernigan said...

You are right Kevin. Thanks for catching my mistake. I've corrected it here.

wattie said...

I did a generalization of the problems with only three judges here:

http://www.cphpvb.net/probability/9251-when-to-follow-authority/

wattie said...

You are missing the "0 0 0 0 0" scenario in the article.