# You are on a jury in a murder trial. After a few days of testimony, your probability for the defendant being guilty is .80. Then, at the end of the trial, the prosecution presents a new piece of evidence, just rushed in from the lab. The defendent’s blood type is found to match that of blood found at the scene of the crime, which could only be the blood of the murderer. The particular blood type occurs in 5% of the population. What should be your revised probability for the defendant’s guilt? Solve the problem first by using Bayes’ rule for probabilites, then by using Bayes’ rule for natural frequencies

## To solve this problem, we will start by using Bayes' rule for probabilities and then proceed to use Bayes' rule for natural frequencies.

1. Bayes' Rule for Probabilities:
Bayes' Rule states that the posterior probability of an event A (in this case, the defendant being guilty) can be calculated based on the prior probability of A, the likelihood of the new evidence given A, and the overall likelihood of the new evidence.

Given:
Prior probability (P(G)): 0.80 (probability for the defendant being guilty)
Likelihood of the evidence given guilt (P(E|G)): 1.00 (the blood type matches the blood found at the crime scene)
Overall likelihood of the evidence (P(E)): 0.05 (blood type occurrence in the population)

Applying Bayes' Rule:
P(G|E) = (P(E|G) * P(G)) / P(E)

P(G|E) = (1.00 * 0.80) / 0.05
P(G|E) = 0.80 / 0.05
P(G|E) = 16

Hence, the revised probability for the defendant's guilt using Bayes' rule for probabilities is 16.

2. Bayes' Rule for Natural Frequencies:
Now, let's solve the same problem using Bayes' rule for natural frequencies, which converts the probabilities into natural frequencies for easier understanding.

Given:
Prior frequency (F(G)): 80% (probability for the defendant being guilty)
Likelihood frequency of the evidence given guilt (F(E|G)): 100% (the blood type matches the blood found at the crime scene)
Overall likelihood frequency of the evidence (F(E)): 5% (blood type occurrence in the population)

Applying Bayes' Rule for Natural Frequencies:
F(G|E) = (F(E|G) * F(G)) / F(E)

F(G|E) = (100% * 80%) / 5%
F(G|E) = 80 / 5
F(G|E) = 16

Therefore, the revised probability for the defendant's guilt using Bayes' rule for natural frequencies is also 16.

In both cases, the revised probability for the defendant's guilt is 16. This indicates a strong increase in the probability of guilt after the presentation of the new evidence.