Likely, what you're asking for is what action do you take based on the initial indications and whatever assumptions made.
The answer, however, is not a concise, definitive act. Rather, the correct answer is a PROCESS.
In troubleshooting, say, an electrical system that isn't working correctly, there is a systematic process called "The 7 Step Process". Let's use this to explain. Those seven steps are:
1. Symptom recognition. This is the first sign that something isn't "right". For an electrical system, this may be as simple as a warning light or a failure to operate correctly in some way. In your example, this is an alarm indication that a window is open.
2. Symptom elaboration. This is where you start to gather some more information to elaborate on the extent of the problem. For an electrical system, that might be looking over all the alarm and warning indications, observing how the equipment is otherwise operating, what procedures were being followed at the time of the initial fault. In your example, this might include checking the alarm system panel for fault indications, listening for unusual noises, seeing if the family pet is disturbed by what's happening.
3. Listing of probable faulty functions. This is where, based on your understanding of the system and how it works, you start listing major things that MIGHT be causing the problem. For an electrical system, if everything appears to be operating properly but the system isn't providing the mechanical function it's supposed to be doing (like opening a door), then you know the fault likely lies in the portions of the control circuits which open the door. In your example, you may have a faulty sensor, faulty alarm system, or an actual event which opened a window.
4. Localizing the faulty function. This is where you start actually troubleshooting...doing something to check/test your hypothesis on where the problem is likely located. Is it a window sensor? Is it the alarm panel? Is it an actual open window?
5. Localizing the trouble to a faulty component. This is further troubleshooting...you know what's causing the alarm (say, a window is actually open), now you're looking for what CAUSED the window to open. Was it a child? Was it a burglar?
6. Failure analysis. This is the point where you evaluate a couple things: Does the cause you found actually match the symptoms you have? Is the cause you found the ONLY problem you have, based upon the nature of the cause and the symptoms you know? In your example, say a child opened a window. This obviously matches the alarm symptoms you have. However, is there any OTHER problem you may have which might also be included in the symptoms? Like, did someone else actually enter the house after your child opened the window?
7. Retest requirements. You found the problem...you fixed the problem. Does the system now work normally?
THE WHOLE POINT of all this is to point out that the actions you take are predicated in large part by the details that emerge along the way during the investigative process.
And, like troubleshooting electrical equipment, there are precautions to take as you perform all these things. Electrical equipment requires me to invoke electrical safety measures, operating procedures, coordination with other people, etc. Likewise, you will have to take analogous measures here: coordinate with other people (wife, children, maybe outside agencies through 911), various safety measures (lights, weapon, dog, cell phone, buddy system), invoking various procedures and training along the way.
So...what would I do?
The first thing is start gathering information so I can decide the appropriate action(s) to take.