A three-man jury (using majority rules) has two members each of whom independently has probability p of making the correct decision and a third member who flips a fair coin for each decision.
A one-man jury has probability p of making the correct decision.
Which jury has the better probability of making the correct decision?
Taking "C" for a 'correct' decision and "F" for a 'false' decision, in order to decide correctly, the decisions of each person must be CCF or CFC or FCC where the first two are the people who decide correctly with probability p and the third is the person who flips a coin. Calculate the probability of each of those and add. Is that greater than p? I think you will find that the answer depends on the value of p.