Question

In the year 2054 the government estimates that out of every 1,000,100 citizens 1,000,000 will remain law-abiding, and 100 will eventually commit a crime. The government is deciding whether to rely on the clairvoyant Precogs to stop the criminals before they commit the crime. If a person is going to commit a crime, the Precogs identify that person correctly 99.999% of the time (in other words, the Precogs fail to predict crime only in 0.001% of the cases). If a person is not going to commit a crime, the Precogs identify that person as law-abiding 99.99% of the times (in other words, the Precogs convict innocent people only in 0.01% of the cases). What is the probability that a person flagged as a future criminal by the Precogs will actually commit a crime

Answer 1

0.9091 = 90.91% probability that a person flagged as a future criminal by the Precogs will actually commit a crime

Explanation:

Bayes Theorem:

Two events, A and B.

`P(B|A) = (P(B)*P(A|B))/(P(A))`

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Flagged as future criminal

Event B: Commits a crime.

In the year 2054 the government estimates that out of every 1,000,100 citizens 1,000,000 will remain law-abiding, and 100 will eventually commit a crime.

This means that
`P(B) = (100)/(1000100) = 0.00009999`

If a person is going to commit a crime, the Precogs identify that person correctly 99.999% of the time

This means that
`P(A|B) = 0.99999`

Probability of being identified as a criminal:

0.99999 out of 0.00009999(identified and commits crime).

(1-0.99999) out of (1-0.00009999) (identifed and does not commits crime). So

`P(A) = 0.99999*0.00009999 + (1-0.99999)*(1-0.00009999) = 0.000109988`

What is the probability that a person flagged as a future criminal by the Precogs will actually commit a crime

`P(B|A) = (P(B)*P(A|B))/(P(A)) = (0.00009999*0.99999)/(0.000109988) = 0.9091`

0.9091 = 90.91% probability that a person flagged as a future criminal by the Precogs will actually commit a crime