So i was working my way through the "Probability Applications" course when the last problem of the last quiz raised a question. How do i discern between conditional probability and probability of intersection, $P(A\mid B)$ and $P(A\cap B)$? When i first read the problem i thought that i needed to calculate the probability of the following events:

- probability of a process being in control
**given**investigation was triggered - probability of a process not being in control
**given**investigation was not triggered.

I managed to work out a solution with this in mind and it was rather plausible. But when i checked the author's solution there were calculations of the probability of the following events:

- probability of a process being in control
**and**investigation was triggered - probability of a process not being in control
**and**investigation was not triggered.

But why? This just bothers me and it seems that i'm missing something important here.

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## Comments

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TopNewestWe need to find the expected value of the savings the control chart. There are four cases, and we need to weight the payoff of each of these by their probabilities. Remember from the end of the problem: there's a 5% chance that a process is out of control.

Giventhat, there is a 99% chance of triggering. So, there is a 5% * 99% = 4.95% chance of control and triggered.We can repeat this for the other scenarios and go from there.

In short, you need to use the fact that $P(A \cap B) = P(B) \cdot P(A|B)$ here, combining the information about a 5% chance of a process being out of control, and then using the conditional probabilities to find the total joint probabilities.

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