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The issue with any test is that it cannot serve two masters. Tests are evaluated by their sensitivity and specificity.
The more sensitive you make a test, you make it less likely to miss someone (false negative), but you run the risk of identifying people as having a disease when they really don't (false positive). With testing sensitivity, you want to identify the true positives.
The more specific a test, the less likely you are to misidentify someone as having a disease when they don't (false positive), but if the test is too specific, you might miss people who have the disease (false negative).
It helps to think of them like a guard dog. An overly sensitive test will be like a dog that barks at every sound, no matter how minor. Will they identify an intruder this way? Yes, but they will also bark at a lot of things that aren't. An overly specific test will be like a dog that doesn't bark until they see someone with a gun in your bedroom. Will the dog not wake you up multiple times per night? Yes. But they also won't alarm you until it is too late.
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