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 probability p of making the correct decision. Which jury has the better probability of making the correct decision?
Solution:
The two juries have the same chance of a correct decision. In the three-man jury, the two serious jurors agree on the correct decision in the fraction p \times p = p^2 of the cases, and for these cases the vote of the joker with the coin does not matter. In the other correct decisions by the three-man jury, the serious jurors vote oppositely, and the joker votes with the “correct” juror. The chance that the serious jurors split is p\left( 1-p \right)+\left( 1-p \right)p or 2p\left( 1-p \right). Halve this because the coin favors the correct side half the time. Finally, the total probability of a correct decision by the three-man jury is p^{2}+p\left( 1-p \right) =p^{2}+p-p^{2}=p, which is identical with the probability given for the one-man jury.
The two options have equal probability of making the correct decision.