Question

Suppose jurors make the right decisions about guilt and innocence 90% of the time and that 70% of all defendants are truly guilty. What is the chance of a convicted (found guilty) defendant being actually innocent?

You can express your answer as a fraction, decimal, or a percentage. If your answer is in decimal, make sure to include at least 3 NON-ZERO digits after the decimal point. If you answer is a percentage, make sure to include a % sign in your answer.

Answer 1

The chance of a convicted defendant being actually innocent can be calculated using Bayes' Theorem. The probability of innocence given conviction is approximately 0.04918 or 4.92%.

Step-by-step explanation:

To find the chance of a convicted defendant being actually innocent, we can use Bayes' Theorem. Let's denote:

P(G) = Probability of guilt = 0.7

P(I) = Probability of innocence = 1 - P(G) = 0.3

P(C|G) = Probability of conviction given guilt = 0.9

P(C|I) = Probability of conviction given innocence (false positive) = 1 - P(C|G) = 0.1

We want to find P(I|C), the probability of innocence given conviction. Using Bayes' Theorem:

P(I|C) = (P(C|I) * P(I)) / (P(C|G) * P(G) + P(C|I) * P(I))

Substituting the values:

P(I|C) = (0.1 * 0.3) / (0.9 * 0.7 + 0.1 * 0.3) = 0.03 / 0.61 ≈ 0.04918

Therefore, the chance of a convicted defendant being actually innocent is approximately 0.04918 or 4.92%.