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?
The table lists the possibilities and probabilites to receive a correct decision.
|Juror 1||Juror 2||Juror 3 (flips)||Majority Decision||Probability|
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:
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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)