Question

Suppose 97% of the population are law abiding citizens, and the remaining citizens break the law.If an innocent person is arrested, they have a 10% chance of confessing, whereas a guilty personconfesses 45% of the time. What is the probability that a person who confessed to a crime is guilty

Answer 1

12.22% probability that a person who confessed to a crime is guilty

Explanation:

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question, we have that:

Event A: Confessing

Event B: Being guilty

Probability of confessing:

10% of 97%(non-guilty) or 45% of 3%(guilty). So

`P(A) = 0.1*0.97 + 0.45*0.03 = 0.1105`

Confessing and being guilty:

3% are guilty, and of those, 45% confess. So

`P(A \cap B) = 0.03*0.45 = 0.0135`

What is the probability that a person who confessed to a crime is guilty?

`P(B|A) = (P(A \cap B))/(P(A)) = (0.0135)/(0.1105) = 0.1222`

12.22% probability that a person who confessed to a crime is guilty