Consider a jury trial in which it takes 8 of the 12 jurors to convict
the defendant; that is, in order for the defendant to be convicted, at
least 8 of the jurors must vote him guilty. If we assume that jurors act
independently and that whether or not the the defendant is guilty; each
makes the right decision with probability θ, what is the the probability
that the jury renders a correct decision?

To determine the probability that the jury renders a correct decision, we need to consider two scenarios:

1. **The defendant is guilty** (and the jury convicts him).
2. **The defendant is not guilty** (and the jury acquits him).

### Assumptions:
- Each juror makes the correct decision with probability \(\theta\).
- There are 12 jurors.
- At least 8 jurors must vote "guilty" to convict the defendant.

Let's define \(X\) as the number of jurors who vote "guilty."

### 1. Probability of Conviction Given Guilt
If the defendant is guilty, then each juror votes "guilty" with probability \(\theta\).

\(X\) follows a binomial distribution:

\[
X \sim \text{Binomial}(n=12, p=\theta)
\]

The probability that the jury convicts the defendant, given that he is guilty, is the probability that at least 8 jurors vote "guilty":

\[
P(\text{Conviction} | \text{Guilty}) = P(X \geq 8)
\]

This can be expressed as:

\[
P(X \geq 8) = \sum_{k=8}^{12} \binom{12}{k} \theta^k (1-\theta)^{12-k}
\]

### 2. Probability of Acquittal Given Innocence
If the defendant is innocent, then each juror votes "not guilty" with probability \(\theta\).

Again, \(X\) follows a binomial distribution, but this time the probability of voting "guilty" is \(1-\theta\):

\[
X \sim \text{Binomial}(n=12, p=1-\theta)
\]

The probability that the jury acquits the defendant, given that he is innocent, is the probability that fewer than 8 jurors vote "guilty":

\[
P(\text{Acquittal} | \text{Innocent}) = P(X < 8)
\]

This can be expressed as:

\[
P(X < 8) = \sum_{k=0}^{7} \binom{12}{k} (1-\theta)^k \theta^{12-k}
\]

### 3. Probability of a Correct Decision
Assuming the defendant is guilty with probability \(p\) and innocent with probability \(1-p\), the overall probability that the jury renders a correct decision is:

\[
P(\text{Correct}) = p \cdot P(\text{Conviction} | \text{Guilty}) + (1-p) \cdot P(\text{Acquittal} | \text{Innocent})
\]

Substituting the expressions derived earlier:

\[
P(\text{Correct}) = p \sum_{k=8}^{12} \binom{12}{k} \theta^k (1-\theta)^{12-k} + (1-p) \sum_{k=0}^{7} \binom{12}{k} (1-\theta)^k \theta^{12-k}
\]

This formula gives the probability that the jury makes a correct decision, either by convicting a guilty defendant or acquitting an innocent one.

### Final Formula
\[
P(\text{Correct}) = p \sum_{k=8}^{12} \binom{12}{k} \theta^k (1-\theta)^{12-k} + (1-p) \sum_{k=0}^{7} \binom{12}{k} (1-\theta)^k \theta^{12-k}
\]

This equation considers both the correct conviction of a guilty defendant and the correct acquittal of an innocent defendant, weighted by their respective probabilities.

Do you want details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How would the probability of a correct decision change if the jury size was increased to 15 jurors?
2. What is the impact of changing the threshold of conviction (e.g., from 8 to 9 jurors) on the probability of a correct decision?
3. How does the assumption of independent decisions by jurors affect the model's accuracy in real-life jury trials?
4. What would happen to the probability of a correct decision if the jurors had varying probabilities of making the correct decision?
5. How can this model be adapted to account for bias or influence among jurors?

**Tip:** The binomial distribution is a fundamental concept in probability theory, frequently used to model scenarios with two possible outcomes, such as success or failure.

Explore the probability that a jury makes a correct decision in a trial scenario where jurors independently decide on the guilt of a defendant. Learn how probability theory and binomial distribution are applied to calculate the likelihood of a correct jury decision.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation