Definitely not a stupid question.
Yes, the management pack can raise notifications when a pool member goes down. The health state of the pool member will change to signify its changed state and the reason. Also, the pool and any associated objects will be marked as unhealthy if the Big-IP is marked as such also ~ e.g. pool marked down by child pool member, or the reverse.
The management pack can also raise notifications for more granular performance-based health. For example, if you decide that 0 total connections on a pool member is unexpected and undesirable, you can create a threshold trigger which will mark the pool member unhealthy when its total connections is 0 or some low number. The threshold will change to red when the connections goes to 0, and return to green when it rises above the threshold.
You can use the built-in Health Explorer in the UI Console (Opsmgr) or PowerShell to identify and handle health state changes.
Thanks for your question, and let us know if you'd like help setting up overrides or reviewing health state.
~Stephen