The probability of a security breach in a network is 5%. A security system detects breaches with 99% accuracy but has a 1% false positive rate. What is the probability that a detected breach is a false alarm?
Question
The probability of a security breach in a network is 5%. A security system detects breaches with 99% accuracy but has a 1% false positive rate. What is the probability that a detected breach is a false alarm?
Solution
To find the probability that a detected breach is a false alarm, we can use Bayes' theorem.
Let's define the events: A: Security breach occurs B: Security system detects a breach
We are given the following probabilities: P(A) = 0.05 (probability of a security breach) P(B|A) = 0.99 (probability that the security system detects a breach given that a breach has occurred) P(B|not A) = 0.01 (probability that the security system detects a breach given that no breach has occurred)
We want to find P(not A|B), which is the probability that a detected breach is a false alarm.
Using Bayes' theorem, we have: P(not A|B) = (P(B|not A) * P(not A)) / P(B)
To calculate P(B), we can use the law of total probability: P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
Substituting the given values, we have: P(B) = 0.99 * 0.05 + 0.01 * (1 - 0.05)
Simplifying, we get: P(B) = 0.0495 + 0.0095 P(B) = 0.059
Now we can substitute the values into Bayes' theorem: P(not A|B) = (0.01 * (1 - 0.05)) / 0.059
Simplifying, we get: P(not A|B) = 0.0095 / 0.059
Calculating this, we find: P(not A|B) ≈ 0.161
Therefore, the probability that a detected breach is a false alarm is approximately 0.161 or 16.1%.
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