Question

The judges of County X try thousands of cases per year. Although in a big majority of the cases disposed the verdict stands as rendered, some cases are appealed. Of those appealed, some are reversed. Because appeals are often made as a result of mistakes by the judges, you want to determine which judges are doing a good job and which ones are making too many mistakes. The attached Excel file has the results of 182,908 disposed cases over a three year period by the 38 judges in various courts of County X. Two of the judges (Judge 3 and Judge 4) did not serve in the same court for the entire three-year period. Using your knowledge of probability and conditional probability you will make an analysis to decide a ranking of judges. You will also analyze the likelihood of appeal and reversal for cases handled by different courts. Using Excel to calculate the following probabilities for each judge. Use the attached Excel file to fill in these numbers on the tables for each court. Each court is given on a separate tab. The probability of a case being appealed for each judge. The probability of a case being reversed for each judge. The probability of a reversal given an appeal for each judge. Probability of cases being appealed and reversed in the three courts. Rank the judges within each court.

Domestic Relations Court
Judge	Total Cases Disposed	Appealed Cases	Reversed Cases	Probability of Appeal	Probability of Reversal	Conditional Probability of Reversal Given Appeal	Rank of Judge
Judge 17	2,729	7	1
Judge 3	6,001	19	4
Judge 18	8,799	48	9
Judge 19	13,970	32	3
Total	31,499	106	17

Answer 1

Judge	Total Cases Disposed	Appealed Cases	Reversed Cases	Probability of Appeal	Probability of Reversal	Conditional Probability of Reversal Given Appeal	Rank of Judge
Judge 3	6,001	19	4	0.003166139	0.210526316	0.210526316	4
Judge 17	2,729	7	1	0.002565042	0.142857143	0.142857143	2
Judge 18	8,799	48	9	0.005455165	0.1875	0.1875	3
Judge 19	13,970	32	3	0.002290623	0.09375	0.09375	1
Total	31,499	106	17

where A is the set of cases appealed and R is a set of cases which are reversed from the set of the appealed cases

a judge is well performed when the no of the reversed cases are less from those of the appealed cases

which is nothing but the probability of reversal given appealed must be less

we can also choose no .of reversal cases as a parameter to give the rankings but no of reversed cases given appealed will be more appropriate(even though his judgement was appealed if the same judgement given by him is given proves his strength to give the right verdict)

remark: here the judge's judgement doesn't depend on the court. hence one need not consider the fact that Two of the judges (Judge 3 and Judge 4) did not serve in the same court for the entire three-year period