The denominator changes the answer
Two questions can sound almost identical while asking for different probabilities: “How often does an alarm sound during a fire?” and “How often is there a fire when the alarm sounds?” The first measures how well the alarm detects fires. The second asks how trustworthy a sounding alarm is. Neither answer can be inferred from the other alone.
The difference is the size of the group each question starts with. Fires may be rare, while nonfires are common. Even a system that catches nearly every fire can produce many false alarms if it mistakenly alerts on a small fraction of the vastly larger number of nonfires.
A hypothetical count makes it visible
Suppose, just for illustration, that a building has 1,000 occasions in which a fire might occur, and only one involves a fire. An alarm catches that fire, but also sounds falsely on 1% of nonfire occasions. That means roughly one genuine alarm and ten false ones. So only about one in eleven alarms would signal a fire—even though the alarm caught the fire.
The key is to count all alarms, not just ask whether the alarm is good at detecting fires. This distinction is often called the base-rate problem: the underlying frequency of an event matters when interpreting evidence about it.
Why intuition slips
People tend to focus on the vivid signal—the alarm—and overlook how many ordinary, fire-free occasions could have produced it. The same mistake can distort judgments about a screening result, a warning system, or a suspicious clue. A useful follow-up is: “Out of everyone or every situation that could trigger this signal, how many are actually cases?”
That question turns an alarming sound into a more grounded estimate of what it means.
