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Answer :
Final answer:
The probability that a person identified as a future terrorist by the surveillance system actually is one is approximately 0.325% when the system's accuracy and the population statistics are taken into account and calculated using Bayes' theorem.
Explanation:
The student is asking about the probability that someone identified as a future terrorist by a surveillance system actually is a future terrorist, given that the system has a 98% chance of correctly identifying a future terrorist and a 99.9% chance of correctly identifying someone who is not a future terrorist. In a population of 300 million people, with 1000 actual future terrorists, if someone is flagged by the system, we need to calculate the probability that this person is indeed a future terrorist (this is known as the Positive Predictive Value, or PPV).
We can use Bayes' theorem to solve this problem:
- True Positive (TP): System correctly identifies a terrorist = 98% of 1000 = 980.
- False Positive (FP): System incorrectly identifies a non-terrorist as a terrorist = 0.1% of (300 million - 1000) = 299,999.
- PPV = TP / (TP + FP).
- PPV = 980 / (980 + 299,999) = 0.003253341.
Therefore, if someone is identified as a future terrorist, the probability that they actually are a future terrorist is approximately 0.325% when rounded to six decimal places.
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Answer:
Given that a person was identified as a future terrorist, there is a 2.8544% probability that he/she actually is a future terrorist.
Step-by-step explanation:
There are 1000 future terrorists in a population of 300,000,000. So the probability that a randomly selected person in this population is a terrorist is:
[tex]P = \frac{1,000}{300,000,000} = 0.000003 = 0.0003%[/tex]
So, we have these following probabilities:
A 99.9997% probability that a randomly chosen person is not a terrorist.
A 0.0003% probability that a randomly chosen person is a terrorist.
A 98% probability that a future terrorist is correctly identified
A 99.9% chance of correctly identifying someone who is not a future terrorist. This also means that there is a 0.01% probability of someone who is not a terrorist being identified as one.
This can be formulated as the following problem:
What is the probability of B happening, knowing that A has happened.
It can be calculated by the following formula
[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]
Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.
Here we have:
What is the probability that the person is a terrorist, given that she was identified as a terrorist.
P(B) is the probability that the person is a terrorist. So [tex]P(B) = 0.000003[/tex]
P(A/B) is the probability that the person was identified as a terrorist, given that she is a terrorist. The problem states that the system has a 98% chance of correctly identifying a future terrorist, so [tex]P(A/B) = 0.98[/tex]
P(A) is the probability of a person being a identified as a terrorist. So
[tex]P(A) = P_{1} + P_{2}[/tex]
[tex]P_{1}[/tex] is the probability that a person is a terrorist and was identified as one. So:
[tex]P_{1} = 0.000003*0.98 = 0.00000294[/tex]
[tex]P_{1}[/tex] is the probability that a person is not a terrorist and, but was identified as one. So:
[tex]P_{2} = 0.999997*0.0001 = 0.0000999997[/tex]
So
[tex]P(A) = P_{1} + P_{2} = 0.00000294 + 0.0000999997 = 0.000103[/tex]
The answer is:
[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.000003*0.98}{0.000103} = 0.028544[/tex]
Given that a person was identified as a future terrorist, there is a 2.8544% probability that he/she actually is a future terrorist.
