What are the practical benefits of having an intuitive understanding of Bayes' Theorem? If it helps, please name an example of how it impacted your day today

I work in tech support (pretty advanced, i.e. I'm routinely dragged into conference calls on 5 minutes notice with 10 people in panic mode because some database cluster is down). Here's a standard situation: "All queries are slow. There are some errors in the log saying something about packets dropped.". So, do I go and investigate all network cards on these 50 machines to see if the firmware is up to date, or do I look for something else? I see people picking the first option all the time. There are error messages, so we have evidence, and that must be it, right? But I have prior knowledge: it's almost never the damn network, so I just ignore that outright, and only come back to it if more plausible causes can be excluded.

Bayes gives me a formal assurance that I'm right to reason this way. I don't really need it quantitatively - just repeating "Base rate fallacy, base rate fallacy" to myself gets me in the right direction - but it's nice to know that there's an exact justification for what I'm doing. Another way would be to learn tons of little heuristics ("No. It's not a compiler bug.", "No. There's not a mistake in this statewide math exam you're taking"), but it's great to look at the underlying principle.