Question

Hamilton County judges try thousands of cases per year. In an overwhelming majority of the cases disposed, the verdict stands as rendered. However, some cases are appealed, and of those appealed, some of the cases are reversed. Kristen DelGuzzi of the Cincinnati Enquirer conducted a study of the cases handled by Hamilton County judges over a three-year period. Shown in Table below are the result for cases handled (disposed) by 4 judges in Domestic Relations Court. The purpose of the newspaper’s study was to evaluate the performance of the judges. Appeals are often the result of mistakes made by judges, and the newspaper wanted to know which judges were doing a good job and which were making too many mistakes. You have been called in to assist in the data analysis. Use your knowledge of probability and conditional probability to help with the ranking of the judges. Prepare a report with your rankings of the judges. Also, include an analysis of the likelihood of appeal and case reversal in the court. At a minimum, your report should include the following:

The probability of reversal given an appeal for each judge.

Rank the judges. State the criteria you used and provide a rationale for your choice.

Domestic Relations Court
Judge	Total Cases Disposed	Appealed Cases	Reversed Case
Penelope Cunningham	2,729	7	1
Patrick Dinkelacker	6,001	19	4
Deborah Gaines	8,799	48	9
Ronald Panioto	12,970	32	3
Total	30,499	106	17

Answer 1

Data:

Different columns of the table below denote the different probabilities.

Judge	Total Cases Disposed, T	Appealed Cases, A	Reversed Case, RC	P(A|Judge)	P(RC|Judge)	P(RC|A)
Ronald Panioto	12,970	32	3	0.002467232	0.000231303	0.09375
Penelope Cunningham	2,729	7	1	0.002565042	0.000366435	0.142857143
Patrick Dinkelacker	6,001	19	4	0.003166139	0.000666556	0.210526316
Deborah Gaines	8,799	48	9	0.005455165	0.001022844	0.1875

Likelihoods of appeal and case reversal in the court.

	Total Cases Disposed, T	Appealed Cases, A	Reversed Case, RC	P(A)	P(RC)	P(RC)
Total	30,499	106	17	0.003475524	0.000557395	0.160377358

Criterion for deciding the ranking of the judges: Proportion of cases reversed

Why?

The proportion of cases reversed against a judge actually tells the mistake committed by a judge while delivering a judgement. Appeals against the judgement may not be an accurate matrix as appeals are filed on the basis of doubt and do not confirm if the judgement was actually a mistake. Also, the probability of reversal given an appeal for each judge may tell you the efficiency of getting him caught right but it does not tell his overall efficiency as it may be the case that he may have fouled very less and also was appealed very less and thus making his probability of reversal given an appeal high but his overall efficiency is actually high. So the most appropriate matrix is the proportion of cases reversed against a judge.

Thus, in my opinion, the ranking of judges will be (from best to worst):

Ronald Panioto > Penelope Cunningham > Patrick Dinkelacker >Deborah Gaines

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