Probability of Correct Decision: Single Juror

Probability of Correct Decision: Single Juror

Overview

This lecture works through the single-juror analysis completely and systematically, deriving all relevant probabilities from first principles. The goal is to make sure every quantity is fully understood before we extend the framework to multiple jurors in Lecture 13.

We start from scratch: a blank slate, one juror, one signal, one decision. By the end, we will have a complete accounting of how often this juror makes the right call, how often she makes each type of mistake, and why her accuracy is necessarily better than random guessing.

The Setup, Restated

The state of the world is either $G$ (the defendant is guilty) or $I$ (the defendant is innocent). Both states are equally likely:

$\text{Prob}(G) = \text{Prob}(I) = \frac{1}{2}$

The juror receives a binary signal $s \in \{g, i\}$, where $g$ suggests guilt and $i$ suggests innocence. The signal has accuracy $p > \frac{1}{2}$:

$\text{Prob}(g \mid G) = p, \qquad \text{Prob}(i \mid G) = 1-p$ $\text{Prob}(i \mid I) = p, \qquad \text{Prob}(g \mid I) = 1-p$

The juror has a threshold of doubt $q \in (0,1)$. She convicts if $\text{Prob}(G \mid s) \geq q$ and acquits if $\text{Prob}(G \mid s) < q$.

From Lecture 10, the posteriors are $\text{Prob}(G \mid g) = p$ and $\text{Prob}(G \mid i) = 1-p$.

We maintain the assumption from Lecture 11 that $1-p < q \leq p$, which ensures the juror follows her signal: she convicts after $g$ and acquits after $i$.

A Complete Accounting: All Four Outcomes

The juror's decision process produces one of four outcomes, depending on the true state and the signal received. Let us enumerate them all and compute each probability.

Outcome 1: State is $I$, signal is $i$. The defendant is innocent and the signal correctly suggests innocence. The juror acquits. This is a correct decision — acquitting the innocent.

The probability of this outcome is: $\text{Prob}(I) \cdot \text{Prob}(i \mid I) = \frac{1}{2} \cdot p = \frac{p}{2}$

Outcome 2: State is $I$, signal is $g$. The defendant is innocent but the signal misleadingly suggests guilt. The juror convicts. This is a mistake — wrongfully convicting an innocent person.

The probability of this outcome is: $\text{Prob}(I) \cdot \text{Prob}(g \mid I) = \frac{1}{2} \cdot (1-p) = \frac{1-p}{2}$

Outcome 3: State is $G$, signal is $g$. The defendant is guilty and the signal correctly suggests guilt. The juror convicts. This is a correct decision — convicting the guilty.

The probability of this outcome is: $\text{Prob}(G) \cdot \text{Prob}(g \mid G) = \frac{1}{2} \cdot p = \frac{p}{2}$

Outcome 4: State is $G$, signal is $i$. The defendant is guilty but the signal misleadingly suggests innocence. The juror acquits. This is a mistake — wrongfully acquitting a guilty person.

The probability of this outcome is: $\text{Prob}(G) \cdot \text{Prob}(i \mid G) = \frac{1}{2} \cdot (1-p) = \frac{1-p}{2}$

Notice that the four probabilities sum to: $\frac{p}{2} + \frac{1-p}{2} + \frac{p}{2} + \frac{1-p}{2} = \frac{p + 1-p + p + 1-p}{2} = \frac{2}{2} = 1$

This is a complete probability accounting — we have covered all possibilities.

The Six Key Probabilities

From the four joint outcomes, we can extract six summary probabilities. Three characterize the juror's accuracy conditional on the true state; three summarize overall performance.

Conditional on the state:

The probability of convicting given the defendant is guilty is the probability that the signal is $g$ when the true state is $G$: $\text{Prob}(\text{convict} \mid G) = p$

The probability of acquitting given the defendant is innocent is the probability that the signal is $i$ when the true state is $I$: $\text{Prob}(\text{acquit} \mid I) = p$

The probability of a wrongful conviction — convicting an innocent defendant — is the probability that the signal is $g$ when the true state is $I$: $\text{Prob}(\text{convict} \mid I) = 1-p$

The probability of a wrongful acquittal — acquitting a guilty defendant — is the probability that the signal is $i$ when the true state is $G$: $\text{Prob}(\text{acquit} \mid G) = 1-p$

Overall performance:

The probability of a correct decision is the probability of either (acquitting an innocent) or (convicting a guilty): $\text{Prob}(\text{correct}) = \frac{1}{2} \cdot p + \frac{1}{2} \cdot p = p$

The probability of a mistake is the probability of either (convicting an innocent) or (acquitting a guilty): $\text{Prob}(\text{mistake}) = \frac{1}{2} \cdot (1-p) + \frac{1}{2} \cdot (1-p) = 1-p$

The key identity: $\text{Prob}(\text{correct}) + \text{Prob}(\text{mistake}) = p + (1-p) = 1$. Every possible outcome is either correct or a mistake. These two events are mutually exclusive and exhaustive.

Comparison with Random Guessing

If the juror ignored her signal entirely and made a random decision — convicting with probability $\frac{1}{2}$ and acquitting with probability $\frac{1}{2}$ — her probability of a correct decision would be $\frac{1}{2}$: she would be right half the time regardless of the true state.

A juror who follows her signal achieves:

• $\text{Prob}(\text{correct}) = p > \frac{1}{2}$
• $\text{Prob}(\text{mistake}) = 1-p < \frac{1}{2}$

Every accurate signal strictly outperforms random guessing. The improvement is $(p - \frac{1}{2})$ in terms of the correct-decision probability. A signal with $p = 0.6$ increases accuracy from $50\%$ to $60\%$. A signal with $p = 0.9$ increases it to $90\%$.

Setting Up Multiple Jurors

The single-juror analysis is now complete. But we designed the institution of trial by jury to have multiple jurors precisely because we expect a group to do better than an individual. Why?

The intuition is that when many jurors each observe independent signals, the true state "shines through" in the aggregate. Some jurors will receive misleading signals, but if there are enough jurors and the signals are independent, the majority of signals will point in the correct direction, and a majority vote will yield the right verdict.

To test this intuition formally, we need to extend the model to $N$ jurors, specify a voting rule, and compute the probability that the jury makes the correct collective decision. We do this in Lecture 13.

The setup for the extension is as follows. There are $N$ jurors (odd for simplicity), each receiving an independent signal with accuracy $p > \frac{1}{2}$. Each juror follows her signal (we maintain the assumption $1-p < q \leq p$). The jury applies some social choice function — we will consider simple majority and unanimity — to the $N$ individual votes. We then compute the six key probabilities for the jury as a whole, analogous to the six quantities computed here for a single juror.

The key question is: how does $\text{Prob}(\text{correct})$ depend on $N$ and on the voting rule? The Condorcet Jury Theorem gives a beautiful and surprising answer.