9. a judge hears the following arguments in a murder hearing. the dna test that places the accused at the murder scene has a true positive rate of 90% (i. e. the probability that the test returning positive given that the accused was actually present at the scene is 0.9). similarly, the dna test has a false negative rate of 80% (i. e. the probability that the test returns negative given that the accused was not present at the scene is 0.8). everyone in the town has a equal probability of being at the murder scene, and the town has a population of 10,000. given the fact that the dna test returned a positive result for the accused, what is the probability that the accused was at the murder scene?

0.0004498

Step-by-step explanation:

Let us define the events:

A = The test returns positive.

B = The accused was present.

Since everyone in the town has an equal probability of being at the murder scene, and the town has a population of 10,000

P(B) = 1/10000 = 0.0001

We have that the probability the test returning positive given that the accused was actually present at the scene is 0.9

P(A | B) = 0.9

and the probability that the test returns negative given that the accused was not present at the scene is 0.8

where

are the complements of A and B respectively.

We want to determine the probability that the DNA test returned a positive result given that the accused was at the murder scene, that is, P(B | A).

We know that P(A | B) = 0.9, so

Now, we have

So if we can determine P(A), the result will follow.

By De Morgan's Law

so

Using the formula

and replacing the values we have found

and finally, the desired result is

i agree because i see the other persons thinking is clear