You're missing a crucial parameter, which is how often you think your system goes out of control. You want a high Pr(problem|alert) but also want a high Pr(ale

You're missing a crucial parameter, which is how often you think your system goes out of control. You want a high Pr(problem|alert) but also want a high Pr(alert|problem), the trade-off you choose between them depends on how often you expect problems to occur. If they are rare then you want false positives to be rare. If problems are frequent then false positives can happen more often without affecting Pr(problem|alert) so much.