Question

Suppose you wanted to evaluate the performance of the three judges in Smallville, Texas: Judge Adams, Judge Brown, and Judge Carter. Over a three-year period in Smallville, Judge Adams saw 26% of the cases, Judge Brown saw 34% of the cases, and Judge Carter saw the remainder of the cases. 3% of Judge Adams’ cases were appealed, 6% of Judge Brown’s cases were appealed, and 9% of Judge Carter’s cases were appealed. Given the judge in a case from this three-year period was not Judge Brown, what is the probability the case was not appealed?

Answer 1

Let A, B and C be the event that the case is with Judge Adams, Judge Brown and Judge Carter respectively.

Let ~A, ~B and ~C be the event that the case is not with Judge Adams, Judge Brown and Judge Carter respectively.

Let Z and ~Z be the event that the case is appealed and not appealed respectively.

Given,

P(A) = 0.26, P(B) = 0.34 and P(C) = 1 - (0.26 + 0.34) = 0.4

P(Z | A) = 0.03 , P(Z | B) = 0.06 and P(Z | C) = 0.09

By law of total probability,

P(Z) = P(Z | A) P(A) +  P(Z | B) P(B) +  P(Z | C) P(C)

= 0.03 * 0.26 + 0.06 * 0.34 + 0.09 * 0.4 = 0.0642

Given the judge in a case from this three-year period was not Judge Brown, what is the probability the case was not appealed

= P(~Z | ~B)

We know that,

P(A | ~B)= (P(A) - P(A | B) P(B)) /( 1- P(B))

and P( ~A | B) = 1− P( A | B)

Thus,

P(~Z | ~B) = 1 - P(Z | ~B)

= 1 - (P(Z) - P(Z | B) P(B)) /( 1- P(B))

= 1 - (0.0642 - 0.06 * 0.34) / (1 - 0.34)

= 1 - 0.06636364

= 0.9336