The Flippant Juror
20 Jul 2015
A threeman jury has two members each of whom independently has probability of making the correct decision and a third member who flips a coin for each decision (majority rules) A oneman jury has probability of making the correct decision. Which jury has the better probability of making the correct decision? ^{1}
Solution
The winning outcomes for the first scenario (three juris) are:
outcome  probability 

Hence, the probability of a correct decision is given by:
The two scenarios are strictly equivalent.
Intuition
Half the times the flippant member agrees with the first member. In these cases the conditional probability of a correct decision is . The other half it agrees with the second member, where the conditional probability is also , thus .

This is problem 3 of Frederick Mosteller’s “Fifty Challenging Problems in Probability”. ↩