Answer:
1.236 × 10^(-3)
Step-by-step explanation:
Let A be the event that the person is a future terrorist
Let B the event that the person is identified as a terrorist
We are told that there are 1,000 future terrorists in a population of 400 million. Thus, the Probability that the person is a terrorist is;
P(A) = 1000/400000000
P(A) = 0.0000025
P(A') = 1 - P(A)
P(A') = 1 - 0.0000025
P(A') = 0.9999975
We are told that the system has a 99% chance of correctly identifying a future terrorist. Thus; P(B|A) = 0.99
Thus, P(B'|A) = 1 - P(B|A)
P(B'|A) = 1 - 0.99
P(B'|A) = 0.01
We are told that there is a 99.8% chance of correctly identifying someone who is not a future terrorist. Thus; P(B'|A') = 0.998
Hence: P(B|A') = 1 - P(B'|A')
P(B|A') = 1 - 0.998
P(B|A') = 0.002
We want to find the probability that someone who is identified as a terrorist, is actually a future terrorist. This is represented by: P(A|B)
We can find it from bayes theorem as follows;
P(A|B) = [P(B|A) × P(A)]/[(P(B|A) × P(A)) + (P(B|A') × P(A')]
Plugging in the relevant values;
P(A|B) = [0.99 × 0.0000025]/[(0.99 × 0.0000025) + (0.002 × 0.9999975)]
P(A|B) = 0.00123597357 = 1.236 × 10^(-3)
