# The Flippant Juror

The 3-person jury makes its decisions based on majority rule. Jurors $$A$$ and $$B$$ each have probability $$p$$ of making the right decision, while Juror $$C$$ has a probability of $$\frac{1}{2}$$ of making the right decision. The cases where they make the right decision are as follows (where $$R$$ represents a right decision, and $$W$$ represents a wrong one):

Juror A Juror B Juror C
R R R
R R W
R W R
W R R

Summing the probabilities of these 4 individual cases gives:

$\frac{p^2}{2} + \frac{p^2}{2} + \frac{p(1-p)}{2} + \frac{p(1-p)}{2} = p$

Thus, the 1-person and 3-person juries both have the same probability of making the right decision.