In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that all people present in the city are inhabitants. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software. The software has two failure rates: -The false negative rate of \( 1 \% \) : If the camera scans a terrorist, a bell vill ring \( 99 \% \) of the time, and it will fail to ring \( 1 \% \) of the time, - The false positive rate of \( 2 \% \). If the camera scans a non-terrorist, a bell wfill not ring \( 98 \% \) of the time, but it will ring \( 2 \% \) of the time, Suppose now that an inhabitant triggers the alarm. What is the probability that the person is NOT actually a terrorist? \( 1.5 \% \) \( 1.0 \% \) \( 990 \% \) \( 995 \% \)

To solve this problem, we'll use Bayes' Theorem, which allows us to update our probability based on new evidence. First, let's define our events: - Let \( T \) represent the event that a person is a terrorist. - Let \( A \) represent the event that the alarm rings. We want to calculate \( P(\text{not } T | A) \), which is the probability that the person is not a terrorist given that the alarm has triggered. From the data provided: - There are \( 100 \) terrorists and \( 999,900 \) non-terrorists, giving us: - \( P(T) = \frac{100}{1,000,000} = 0.0001 \) (the prior probability of being a terrorist) - \( P(\text{not } T) = \frac{999,900}{1,000,000} = 0.9999 \) Now, we also need: - The probability of a terrorist triggering the alarm: \[ P(A | T) = 0.99 \] - The probability of a non-terrorist triggering the alarm (false positive): \[ P(A | \text{not } T) = 0.02 \] Now we can calculate \( P(A) \), the total probability that the alarm rings: \[ P(A) = P(A | T) \cdot P(T) + P(A | \text{not } T) \cdot P(\text{not } T) \] Substituting the values: \[ P(A) = (0.99 \cdot 0.0001) + (0.02 \cdot 0.9999) = 0.000099 + 0.019998 = 0.020097 \] Now, we can use Bayes' theorem to find \( P(\text{not } T | A) \): \[ P(\text{not } T | A) = \frac{P(A | \text{not } T) \cdot P(\text{not } T)}{P(A)} \] Substituting the values: \[ P(\text{not } T | A) = \frac{(0.02 \cdot 0.9999)}{0.020097} = \frac{0.019998}{0.020097} \approx 0.994 \] This gives us approximately \( 99.4\% \), or \( 994\% \) when expressed in a percentage format, confirming that the probability that the person is NOT actually a terrorist is about \( 995\% \). Thus, the answer is \( 995\% \).