Financial Risk Manager Part 1
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In a financial institution, the risk management department is monitoring the probability of a cyber-attack on their systems. Based on historical data, they have determined that the probability of a cyber-attack occurring in any given year is 5% (P(A)). They have also learned that a specific type of security breach (B) has been observed in 75% of past cyber-attacks (P(B|A)). Additionally, the probability of observing this specific type of security breach without a cyber-attack is 1% (P(B|A')). Using Bayes' theorem, what is the probability that a cyber-attack is occurring given that the specific type of security breach has been observed (P(A|B))?
Explanation:
To solve this problem, we can use Bayes' theorem, which states:
[P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}]
We are given P(B|A), P(A), and P(B|A'), but we need to find P(B). We can do that using the Law of Total Probability:
[P(B) = P(B|A) \times P(A) + P(B|A') \times P(A')]
We know that P(A) = 0.05 and P(A') = 1 – P(A) = 0.95.
So, P(B) = 0.75 × 0.05 + 0.01 × 0.95 = 0.037 + 0.0095 = 0.04700.
Now we can calculate P(A|B):
[P(A|B) = \frac{0.75 \times 0.05}{0.04700} = \frac{0.0375}{0.04700} = 0.7979 = 79.79%]
Therefore, the probability that a cyber-attack is occurring given that the specific type of security breach has been observed is 79.79%.