Political Science The probability that a three-person jury...

Political Science The probability that a three-person jury will make a correct decision is given by $\begin{aligned} P(\alpha, r, s)=& \alpha\left[3 r^{2}(1-r)+r^{3}\right] \\ &+(1-\alpha)\left[3 s^{2}(1-s)+s^{3}\right] \end{aligned}$ where $0<\alpha<1$ 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. Source: Frontiers of Economics. (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 $\alpha$ in this problem? Should they? What is the maximum probability? (c) Verify your answer for part (b) using calculus. (Hint: There are two critical points. Argue that the maximum value occurs at one of these points.)

Political Science The probability that a three-person jury will make a correct decision is given by
$\begin{aligned} P(\alpha, r, s)=& \alpha\left[3 r^{2}(1-r)+r^{3}\right] \\ &+(1-\alpha)\left[3 s^{2}(1-s)+s^{3}\right] \end{aligned}$
where $0<\alpha<1$ 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. Source: Frontiers of Economics.
(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 $\alpha$ in this problem? Should they? What is the maximum probability?
(c) Verify your answer for part (b) using calculus. (Hint: There are two critical points. Argue that the maximum value occurs at one of these points.)

Key Concepts

Jury Decision Models

This concept involves modeling how a group (in this case a three-person jury) arrives at a decision based on the votes of its individual members. The overall probability of a correct decision is computed by considering the different ways in which a majority (at least two out of three) can agree with the correct decision and then combining these probabilities appropriately. The model shows how individual reliability translates into an aggregate decision-making process.

Binomial Probability Distribution

Although not explicitly labeled as such, the formula for the probability that a majority votes correctly uses terms that come from the binomial distribution. In a three?member jury, the probabilities of exactly two or exactly three jurors voting correctly are given by expressions similar to the binomial coefficients times the individual success probabilities raised to various powers. This method of summing over the number of correct votes is a standard technique in probability theory.

Weighting by Prior Probabilities

The overall probability of a correct verdict is computed by considering two mutually exclusive cases – when the defendant is truly guilty and when he is not – and weighting each case by its prior probability (? for guilt and 1?? for innocence). This combines the likelihood of the facts with the reliability of each juror’s vote in their respective cases, showing a mixture model approach where different scenarios are accounted for in the final decision probability.

Non?Calculus Optimization

Without resorting to calculus, one can reason by common sense that if each juror is perfectly reliable (always casting the correct vote), then the jury as a whole will always make the correct decision. Hence, the maximum value of the probability function occurs when each juror’s probability of making the correct decision (r for voting guilty when guilty and s for voting innocent when innocent) is 1. This intuitive argument shows that the optimum does not depend on the weight ? – regardless of the actual crime probability, perfect juror reliability guarantees a correct majority decision.

Calculus?Based Optimization Verification

To rigorously confirm the intuitive answer, one can differentiate the probability function with respect to the reliability parameters (r and s) and set these derivatives to zero. This procedure yields critical points corresponding to r=1 and s=1 (as well as the trivial lower boundary of 0), and a second derivative test shows that these are indeed maximum points. This calculus-based method verifies that the maximum probability of a correct jury decision is 1 when both r and s are equal to 1, independent of the value of ?.

Transcript

00:01 All right, so here we have this interesting function.

00:06 P of alpha, r, and s is equal to alpha times 3 r squared, times 1 minus r, plus r cute, plus 1 minus alpha, and then 3s squared times 1 minus s squared times 1 minus s plus 1 minus s, plus s cubed.

00:35 And now, p is the probability that a jury of three people will make a correct decision.

00:47 Alpha is the probability that the person is actually guilty.

00:52 R is the probability that a given jury member will vote guilty when the defendant is indeed guilty.

01:00 S is the probability that a given jury member will vote innocent when the defendant is indeed innocent.

01:06 So part a asks us just to evaluate a couple of values.

01:12 So p of 0 .9, 0 .5, 0 .6.

01:25 Okay, so it gives us 0 .5, 148, and then p.

01:32 0 .1, 0 .8, 0 .4 .4.

01:40 And that's 0 .4064.

01:47 So here, we have a situation where there's a high likelihood the defendant is guilty, and there's a little bit of bias to say that the defendant is innocent.

02:06 They'll say that the defendant is innocent if he's actually innocent, whereas here, the defendant has a low, so there's a low probability that defendant is guilty, and then there's a bias towards saying that the defendant is guilty if he's actually guilty.

02:25 So in this situation, under these parameters, so this jury is less likely to be correct.

02:39 Okay.

02:40 And so b says, okay, well, what is the actual maximum? just using common sense.

02:49 Well, if the jury has some type of perfect knowledge, so there's a hundred percent chance that they will accurately say the defendant is guilty if they're guilty, and 100 percent the jury will say the defendant is innocent if they're innocent, then of course there's a hundred percent chance that they're going to correctly get the correct verdict.

03:15 So what that's saying is that if, well, regardless of whether the defendant is innocent or guilty, if i just said r &s to be one, i'm going to get one.

03:28 Okay, so i have a perfect jury somehow.

03:30 It doesn't matter the probability or how likely it is the defendant is guilty or innocent.

03:37 The jury is going to get it right.

03:39 So that's sort of like the heuristic of why the maximum of this function is one.

03:44 And it's achieved when r is 1 and s is 1.

03:49 And actually we can verify that just by plugging in r equals 1 and s equals 1 will get 1...

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