aez-notes
The Flippant Juror
A three-man jury has two members each of whom independently has probability p of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has a probability p of making a correct decision. Which jury has the better probability of making the correct decision?
The possible outcomes can be enumerated as follows
| Juror 1 | Juror 2 | Juror 3 | Correct decision | Probability |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 | 0 | |
| 0 | 1 | 0 | 0 | |
| 0 | 1 | 1 | 1 | \((1-p) p (1/2)\) |
| 1 | 0 | 0 | 0 | |
| 1 | 0 | 1 | 1 | \(p(1/2)p\) |
| 1 | 1 | 0 | 1 | \(p(1/2)(1-p)\) |
| 1 | 1 | 1 | 1 | \(p(1/2)p\) |
prob_three_correct: p^2 + p*(1-p); prob_one_correct: p; print(is(equal(prob_three_correct, prob_one_correct)));
This prints out "true", telling us that the juries have an equal probability of making the correct decision.