# Probability Problem Number 3 of Frederick Mostellers Book

In 1916 Frederick Mosteller published a collection of problems with solutions in his famous book „Fifty Challenging Problems in Probability„.

Problem Number 3 is called „The Flippand Juror“ and is described with the following lines:

A three-man jury has two members each of whom independently has a 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 a probability p of making the correct decision. Which jury has the better probability of making the right decision?

Three-man jury:

The table lists the possibilities and probabilites to receive a correct decision.

 Juror 1 Juror 2 Juror 3 (flips) Majority Decision Probability C C C C C C W C C W C C W C C C

The single probabilites sum up to:

The three-man jury with the flippand juror has the same probability for correct decisions p as a one man jury with a „real“ juror.

Python code to simulate the problem:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 import numpy as np     p = 0.9 # Probability for the right decision of a real juror pf = 0.5 # Probability for the right decicion of the flippand juror   #judges_correct = [0,0,0] n_sim = int(1e4) # number of simulated jury decicions   jury_correct_count = 0 for i in range(0, n_sim): judges_correct = np.random.uniform(low=0, high=1, size=3) <= [p,p, pf] # boolean array of right/wrong decisions   if sum(judges_correct) >= 2: # majority rules, jury did a right decision jury_correct_count = jury_correct_count + 1 # increment the counter of right jury decicions     p_jury_correct = jury_correct_count / n_sim print(p_jury_correct)