The probability that a three-person jury will make a correct decision is given by

where 0 < a < 1 is the probability that the person is guilty of the crime, r is the probability that a given jury member will vote “guilty” when the defendant is indeed guilty of the crime, and s is the probability that a given jury member will vote “innocent” when the defendant is indeed innocent.

(a) Calculate P(0.9, 0.5, 0.6) and P(0.1, 0.8, 0.4) and interpret your answers.
(b) Using common sense and without using calculus, what value of r and s would maximize the jury’s probability of making the correct verdict? Do these values depend on a in this problem? Should they? What is the maximum probability?
(c) Verify your answer for part (b) using calculus.

Transcribed Image Text:

P(a, r, s) = a[3r²(1 − r) + r³] + (1 - a)[3s²(1 - s) + s³],