A three person jury decides by majority vote There are two j

A three- person jury decides by majority vote. There are two jurors each of whom independently has probability p of making the correct decision, and a third juror who flips a coin for each decision. A one- person jury has probability p of making the correct decision. Which jury has the higher probability of making the correct decision?

Solution

Both juries have the same probability (p) of making the correct decision.
The probability that the first jury makes the right decision is the
probability that either exactly two or all three jurors make the right
decision. If we assume that q=1-p and that the third juror is the coin
flipper then:

P(right, right, wrong) = p^2 /2
P(right, wrong, right) = pq /2
P(wrong, right, right) = pq /2
P(right, right, right) = p^2 /2

which adds up to p^2+pq = p(p+q) = p.