Question

A three-person jury has two members who each have a probability p of making the correct decision in a case. The third member doesnt care and flips a coin for each decision. The ruling is based on a majority vote amongst the jurors

(a) What is the probability that the jury will correctly decide the case?
(b) Suppose two of the jurors quit, one of whom is the juror that doesnt care. Does the rrectly deciding the case i ncrease, decrease, or no at all?

Answer 1

(a) p

(b) the probability does not change at all

Explanation:

(a) Let A and B be the jurors with probability 'p' of making the correct decision, and C be the juror that doesn't care. The case will be correctly decided if any of the following combinations of jurors decide the case correctly:

AB, AC, BC, ABC.

The probability of one of those outcomes occurring is:

`P=(p*p*0.5)+(p*(1-p)*0.5)+(p*(1-p)*0.5)+(p*p*0.5)\\P=p^2+p-p^2\\P=p`

The probability is p.

(b) If two of the juros quit, the probability of correctly deciding the case lies on just one juror that correctly decides with probability 'p'. Therefore, the probability of deciding the case does not change at all