Answer:
0.0748 = 7.48% probability that he is guilty
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
![P(B|A) = \frac{P(A \cap B)}{P(A)}](https://tex.z-dn.net/?f=P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D)
In which
P(B|A) is the probability of event B happening, given that A happened.
is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Tests indicates guilty
Event B: Person is guilty
Probability that the test indicates that the person is guilty:
4/5 = 0.8 of 1/100 = 0.01(person is guilty)
1 - 0.9 = 0.1 of 1 - 0.99(person is not guilty). So
![P(A) = 0.8*0.01 + 0.1*0.99 = 0.107](https://tex.z-dn.net/?f=P%28A%29%20%3D%200.8%2A0.01%20%2B%200.1%2A0.99%20%3D%200.107)
Test indicates guilty and the person is guilty;
0.01 of 0.8. So
![P(A \cap B) = 0.01*0.8 = 0.008](https://tex.z-dn.net/?f=P%28A%20%5Ccap%20B%29%20%3D%200.01%2A0.8%20%3D%200.008)
What is the probability that he is guilty?
![P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.008}{0.107} = 0.0748](https://tex.z-dn.net/?f=P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D%20%3D%20%5Cfrac%7B0.008%7D%7B0.107%7D%20%3D%200.0748)
0.0748 = 7.48% probability that he is guilty