Question

DNA evidence is often used in criminal trials. Suppose a murder has been committed and the perpetrator's blood is found at the crime scene. DNA from the blood is analyzed and it is found that the probability of an innocent person having DNA that matches is 10⁻⁸. Let M be the event 'DNA match' and I be the event 'Innocent'. Therefore, we have P(M|I) 10⁻⁸. What is really of interest is the quantity P(I|M), the probability that a person whose DNA matches is innocent. Calculate this quantity if the community has 1000 people, and therefore the probability that a randomly chosen person is innocent is P(I) 999/1000. You can assume that the probability of a match for the guilty person is 1 (that is, P(M|I) 1). (Round your answer to three decimal places.)

Answer 1

To calculate the probability that a person whose DNA matches is innocent, we can use Bayes' theorem. Plugging in the given values, we find that the probability that a person whose DNA matches is innocent is approximately 0.999.

Step-by-step explanation:

To calculate the probability that a person whose DNA matches is innocent, we can use Bayes' theorem. Bayes' theorem states that the probability of event A given event B is equal to the probability of event B given event A multiplied by the probability of event A, divided by the probability of event B.

In this case, event A is being innocent (I) and event B is a DNA match (M). We are given that P(M|I) = 1 (the probability of a match for the innocent person is 1), P(I) = 999/1000 (the probability that a randomly chosen person is innocent), and P(M|I') = 10-8 (the probability of a match for an innocent person is 10-8).

Using Bayes' theorem, we can calculate P(I|M) as follows:

1. Calculate P(M|I') = 1 - P(M'|I') = 1 - 10-8 = 0.99999999
2. Calculate P(I|M) = (P(M|I) * P(I)) / ((P(M|I) * P(I)) + (P(M|I') * P(I'))) = (1 * (999/1000)) / ((1 * (999/1000)) + (0.99999999 * (1/1000)))
3. Finally, calculate the numerical value of P(I|M).

Plugging in the values, we get:

1. P(I|M) = (1 * (999/1000)) / ((1 * (999/1000)) + (0.99999999 * (1/1000)))
2. P(I|M) ≈ 0.999