Problem 1

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 the probability p of making the correct decision. Which jury has the better probability of making the correct decision (the 3-member jury or the one-member jury)?
Solution.
Let A, B, C be the jurors and let A be the one who flips the coin. Different probabilities can be represented using the tree diagram below:
figure Fig_problem_1.jpg
The probability that the three member jury takes a correct decision is the following sum:
(1)/(2)⋅p⋅p + (1)/(2)⋅p⋅(1 − p) + (1)/(2)⋅(1 − p)⋅p + (1)/(2)⋅(1 − p)⋅(1 − p) = p2 + p(1 − p) = p.
So, we conclude that both juries have the same probabil;ity p.