A woman was accused of killing someone in her town due to a blood type match. The blood type is a rare variant that is present only in 0.1% of the entire population. If the population of her town is 5,000, what are the chances that she is the killer?

To solve this, we need to analyze the situation using **Bayesian probability** since it involves updating the probability of an event (the woman being the killer) given new evidence (the blood type match).

### Step 1: Define the known information
- The rare blood type occurs in **0.1%** of the population, so \( P(\text{blood match}) = 0.001 \).
- The population of the town is **5,000**.
- Let \( P(\text{guilty}) \) be the probability that the woman is the killer.
- Let \( P(\text{not guilty}) = 1 - P(\text{guilty}) \).
- The blood type match could happen whether or not she is guilty.

### Step 2: Probabilities to consider
We need to calculate the **posterior probability** that she is guilty, given that there is a blood type match, using Bayes' Theorem:

\[
P(\text{guilty} \mid \text{blood match}) = \frac{P(\text{blood match} \mid \text{guilty}) \cdot P(\text{guilty})}{P(\text{blood match})}
\]

Where:
- \( P(\text{blood match} \mid \text{guilty}) = 1 \) (If she's guilty, she must have the matching blood type.)
- \( P(\text{guilty}) \) is the prior probability that the woman is guilty before considering the blood type evidence. We assume no prior knowledge of guilt, so \( P(\text{guilty}) \) could be low, e.g., 1 person out of 5,000, or \( P(\text{guilty}) = \frac{1}{5000} \).

### Step 3: Calculate the denominator \( P(\text{blood match}) \)
The total probability of a blood type match can occur in two ways:
1. The woman is guilty and has the matching blood type.
2. The woman is not guilty but still has the matching blood type by chance.

Thus:
\[
P(\text{blood match}) = P(\text{blood match} \mid \text{guilty}) \cdot P(\text{guilty}) + P(\text{blood match} \mid \text{not guilty}) \cdot P(\text{not guilty})
\]

Where:
- \( P(\text{blood match} \mid \text{guilty}) = 1 \)
- \( P(\text{blood match} \mid \text{not guilty}) = 0.001 \) (the chance someone else in town has the same blood type)
- \( P(\text{not guilty}) = 1 - \frac{1}{5000} \approx 0.9998 \)

### Step 4: Plug values into the formula

\[
P(\text{blood match}) = 1 \cdot \frac{1}{5000} + 0.001 \cdot 0.9998
\]
\[
P(\text{blood match}) = \frac{1}{5000} + 0.0009998 \approx 0.0012
\]

### Step 5: Calculate the posterior probability

Now use Bayes' Theorem to find the probability that the woman is guilty given the blood match:

\[
P(\text{guilty} \mid \text{blood match}) = \frac{1 \cdot \frac{1}{5000}}{0.0012} \approx \frac{0.0002}{0.0012} \approx 0.167
\]

So, the probability that she is guilty given the blood match is approximately **16.7%**.

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Would you like more details or further explanations on any part of this process?

Here are some related questions:
1. What is Bayes' Theorem, and why is it useful in probability problems?
2. How do we calculate prior probabilities in real-world situations?
3. What assumptions might affect the accuracy of this Bayesian analysis?
4. Can we improve the probability estimate by considering additional evidence?
5. How would changing the size of the population affect the result?

**Tip:** Bayes' Theorem is especially useful when you need to update probabilities based on new evidence, making it powerful in legal, medical, and other real-world contexts.

Explore how to calculate the probability of guilt using Bayes' Theorem, given a rare blood type match in a population of 5,000. Learn the step-by-step solution, including key probability concepts and formulas, such as conditional probability and total probability. Suitable for college-level students.

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