Question

Over a three year period in Smallsville, Judge Adams saw 37% of the cases, Judge Brown saw 39% of the cases and Judge Carter saw the remainder of the cases. Nine percent of Judge Adams' cases were appealed, 8% of Judge Browns's cases were appealed, and 6% of Judge Carters cases were appealed.

Given a randomly selected case from this three year period was not appealed, what is the probability the judge in the case was not Judge Carter?

Answer 1

Question #15222 Over a three year period in Smallsville, Judge Adams saw 37% of the cases, Judge Brown saw 39% of the cases and Judge Carter saw the remainder of the cases. Nine percent of Judge Adams’ cases were appealed, 8% of Judge Browns’s cases were appealed, and 6% of Judge Carters cases were appealed.

Given a randomly selected case from this three year period was not appealed, what is the probability the judge in the case was not Judge Carter?

Solution. Denote by $A,B,C$ respectively respectively the events that random case was seen by Adams, Brown, Carter and by $NA$ the randomly selected case was not appealed. The condition implies that $P(A)=0.37,P(B=0.39,P(C)=0.24)$ and $P(NA|A)=0.91,P(NA|B)=0.92,P(NA|C)=0.94$. We are to calculate $P(\overline{C}|NA)=1-P(C|NA)$, using Bayesian formula one can get the last equals $1-\frac{P(NA|C)P(C)}{P(NA|A)P(A)+P(NA|B)P(B)+P(NA|C)P(C)}=1-\frac{0.94\cdot 0.24}{0.91\cdot 0.37+0.92\cdot 0.39+0.94\cdot 0.24}\approx 1-0.245=0.755$.

Answer Approximately 0.76.