# Bayes Theorem Question with compound events

I'm very rusty here, so this is probably a lot simpler than I'm trying to make it. Taking an online stats course and encountered this question.... Systems A and B are candidate fault detection systems. If there is a fault, system A raises an alert with a probability of 0.9, and system B raises an alert with a probability of 0.95. If there is no fault, system A raises a (false) alarm with a probability of 0.2, and system B raises a (false) alarm with a probability of 0.1. The two systems operate independently.

Assume that the probability of a fault is 0.4. If both systems raise an alarm, what is the probability there is actually a fault?

So if we let:

• A be the event system A raises an alarm
• B be the event system B raises an alarm
• F be the event there is actually a fault

I can gather that:

• $$P(A|F)=0.9$$
• $$P(B|F)=0.95$$
• $$P(A|F')=0.2$$
• $$P(B|F')=0.1$$

And I believe that $$P(A \cap B|F) = P(A|F) \cdot P(B|F) = (0.9) \cdot (0.95) = 0.855$$ since they're independent.

So I think this is a Bayes Theorem question where we are looking to find $$P(F|A \cap B)$$

But then I start with $$P(F|A \cap B) = \frac{P(F) \cdot P(A \cap B|F)}{P(A \cap B)}$$ and am not sure where to go with that $$P(A \cap B)$$ in the denominator. Is this an application of total probability to find $$P(A \cap B)$$ or am I way off?

The calculation is going to depend on whether you interpret "The two systems operate independently" as conditional independence of fault detection $$P(A \cap B \mid F) = P(A\mid F) \cdot P(B\mid F)$$ and $$P(A \cap B \mid F') = P(A\mid F') \cdot P(B\mid F')$$ even though here they would together be inconsistent with unconditional independence $$P(A \cap B ) = P(A) \cdot P(B)$$. As another issue, in real life "operational independence" is not the same thing as predictive independence.

• $$P(F \mid A \cap B)=\dfrac{P(A \cap B \cap F)}{P(A \cap B)}$$
• $$P(A \cap B) = P(A \cap B \cap F) + P(A \cap B \cap F')$$
• $$P(A \cap B \cap F) = P(F) \cdot P(A \cap B\mid F)$$
• $$P(A \cap B \cap F') = P(F) \cdot P(A \cap B\mid F')$$
• $$P(A \cap B)= P(F) \cdot P(A \cap B\mid F)+ P(F') \cdot P(A \cap B\mid F')$$
• $$P(F \mid A \cap B)=\dfrac{P(F) \cdot P(A \cap B\mid F)}{P(F) \cdot P(A \cap B\mid F)+ P(F') \cdot P(A \cap B\mid F')}$$