The Appeals Court Paradox
The Appeals Court Paradox
A criminal appeals court consists of five judges: A, B, C, D, and E. Consider two cases: in case 1, each judge votes independently, and in case 2, judge E always votes as judge A does. This Demonstration shows the probabilities of various numbers of correct decisions in each case.
The default settings of the Demonstration show an example of the so-called appeals court paradox. In this example, a majority of the votes determines the decision of the appeals court. Also, the probability that judge E makes the correct decision is 0.8. As this probability is smaller than the probability (0.95) that judge A makes the correct decision, it seems useful that judge E should always vote as judge A does; in this way it is assumed that the court arrives at the correct decision with a larger probability. However, the opposite holds: in case 1, the court makes an incorrect decision with probability 0.007055 (this is the probability that we get at most two correct decisions), while in case 2 that probability, 0.012, is almost twice as large as in case 1.