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In a recent federal appeals court case, a special 11-judge panel sat to decide on a certain particular legal issue under certain particular facts. Of the 11 judges, 3 were appointed by political party A, and 8 were appointed by political party B. Of the party-A judges, 2 of 3 sided with the Plaintiff on the issue, and of the party-B judges 6 of 8 sided with the Plaintiff. Now in a later appeal of a different case involving the same legal issue and same exact facts, but where a regular 3-judge panel sat to decide the appeal, these 3 judges decided unanimously (3 of 3) against the Plaintiff. The 3-judge panel was composed of two (2) party-A type judges, and one (1) party-B type judge. The Plaintiff claimed that the 3 judge panel was biased and prejudiced against him. Is there any merit to what the Plaintiff is claiming?

I see this as a binomial type problem, but with different probabilities involved that need to be accounted for – a bivariate type situation.

But can I simplify this problem by averaging the probabilities of the judges voting for or against the Plaintiff? If I can, then this becomes a single-variable type problem and I can just use the good-old binomial formula to determine a z-score and confidence interval.(?)

If my averaging method is okay to do, this is what I come up with:

The probability of a decision against the Plaintiff by a party-A judge is 0.33, and the probability of a decision against the Plaintiff by a party-B judge is 0.25. There were two party-A judges and one party-B judge on the panel, so the probability average of the judges voting against the Plaintiff is (0.33+0.33+0.22)/3 = 0.30.

Thus, I can plug the binomial formula like so --> p=0.30, q=0.70, n=3, and x=3. Really, in such a situation, the binomial formula boils down to 0.30^3 = 0.027 <-- call it 0.03

Now, I can also find the s.d. with these numbers, and I find s.d.= 0.794

Also, the expected (mean) number of votes against the Plaintiff would be 0.33+0.33+0.25 = 0.91

Plugging the z score formula I get z = (x – u)/(s.d.) = (3 – 0.91)/0.794 = 2.63

That is 2.63 s.d. from the mean (of 0.91 total average votes against the Plaintiff by a randomly selected and unbiased 3-judge appellate panel), which translates to a 0.0043 probability that such an occurrence would happen from random chance alone. This would indicate that the judges were, indeed, biased and prejudiced against the Plaintiff. There is “merit” in what the Plaintiff claims.

Okay, so what is the right way to do this problem, and with a confidence interval? (Will be interesting to see the difference between doing it the right way and the way I did it.)

I see this as a binomial type problem, but with different probabilities involved that need to be accounted for – a bivariate type situation.

But can I simplify this problem by averaging the probabilities of the judges voting for or against the Plaintiff? If I can, then this becomes a single-variable type problem and I can just use the good-old binomial formula to determine a z-score and confidence interval.(?)

If my averaging method is okay to do, this is what I come up with:

The probability of a decision against the Plaintiff by a party-A judge is 0.33, and the probability of a decision against the Plaintiff by a party-B judge is 0.25. There were two party-A judges and one party-B judge on the panel, so the probability average of the judges voting against the Plaintiff is (0.33+0.33+0.22)/3 = 0.30.

Thus, I can plug the binomial formula like so --> p=0.30, q=0.70, n=3, and x=3. Really, in such a situation, the binomial formula boils down to 0.30^3 = 0.027 <-- call it 0.03

Now, I can also find the s.d. with these numbers, and I find s.d.= 0.794

Also, the expected (mean) number of votes against the Plaintiff would be 0.33+0.33+0.25 = 0.91

Plugging the z score formula I get z = (x – u)/(s.d.) = (3 – 0.91)/0.794 = 2.63

That is 2.63 s.d. from the mean (of 0.91 total average votes against the Plaintiff by a randomly selected and unbiased 3-judge appellate panel), which translates to a 0.0043 probability that such an occurrence would happen from random chance alone. This would indicate that the judges were, indeed, biased and prejudiced against the Plaintiff. There is “merit” in what the Plaintiff claims.

Okay, so what is the right way to do this problem, and with a confidence interval? (Will be interesting to see the difference between doing it the right way and the way I did it.)

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