Fifty Challenging Problems In Probability
Selected problem famously known as the 3-Man jury.
Consider a 3-man jury: two serious ones and a joker. We’re assuming the probability of each serious juror independently making the correct judgment is p, and the joker makes a random decision (with a probability of 0.5 for being correct since their decision is random, like flipping a coin).
Two Serious Jurors Agree
- Both Correct: Probability is p×p=p².
- Both Incorrect: Probability is (1−p)×(1−p)=(1−p)².
Two Serious Jurors Disagree
- One is correct, and the other is incorrect. The probability for this scenario is 2×p×(1−p), as this can happen in two ways (Juror 1 is correct and Juror 2 is incorrect, or vice versa).
Jury Decision with Joker
- Both Serious Jurors Agree: The decision is already made, irrespective of the joker’s vote. The probability that this decision is correct is p², and the probability that it’s incorrect is (1−p)².
- Serious Jurors Disagree: The decision falls to the joker, effectively a random decision. The probability of a correct decision is 0.5, and incorrect is also 0.5.
Probability of Correct Jury Decision
- When the serious jurors agree, the probability of a correct decision is p².
- When they disagree, the decision is a coin flip (0.5 chance of being correct). The probability of them disagreeing is 2×p×(1−p), so the contribution to the overall probability of a correct decision is 0.5×2×p×(1−p)=p(1−p).
Therefore, the total probability of a correct decision by the 3-person jury is p²+p(1−p) = p.
Compared with One Serious Juror
A single serious juror has a probability p of making the correct decision. So, A 3-man jury with a Joker is just as reasonable as a 1-man jury.
Why do Batman and Gordon need Two-Face?
Without Two-Face (referred to as Joker in this article), the jury (Batman and Gordon) can only make a correct decision when both agree, which has a probability of p².
- With Two-Face: The probability of a correct decision is p.
- Without Two-Face: The probability is p².
Comparing these two probabilities demonstrates that the inclusion of Harvey Dent, aka Two-Face, in the jury increases the overall probability of making a correct decision. The addition of Two-Face’s random decision-making paradoxically, enhances the jury’s effectiveness in scenarios where Batman and Gordon disagree.
It's fascinating how a chance element can enhance the results of group decision-making.