Imagine you test positive for a test that predicts that you have a rare disease with 99% accuracy. You would worry right? You might think this means you almost certainly have the disease, with only a 1% chance of not having the disease. This seems very intuitive, but is actually completely wrong. If the disease is rare, a positive result from this remarkably sensitive test can still mean that you are almost certainly healthy. This feels impossible at first. How can a test that almost never misses a true case be wrong most of the time when it gives a positive result?
The answer has nothing to do with the machine that runs the test and everything to do with the mathematics behind it. When a test is used on a group of people who are very unlikely to have the disease, the number of false positives produced by healthy individuals can easily outnumber the true positives produced by the few people who are actually sick. This is a situation where statistical intuition fails and where Bayes rule becomes essential for interpreting what a test result really means.
Imagine a disease that affects only one in one thousand people. We use a test that correctly identifies 99% of true cases and correctly clears 99% of healthy people. It seems perfect. However, if you test positive, the chance that you truly have the disease is not 99%. It is closer to 9%. This surprises almost everyone, including many people who work with data.
The reason is simple once you look at the numbers. In a group of 1000 people, one person really has the disease. The test will almost certainly catch that one person. But there are 999 healthy people in the room. Even with an error rate of only one percent, the test will incorrectly flag around ten of them. As a result we have one true positive and ten false positives. Among those who receive a positive result, only one in eleven (9%) is actually sick.
This example feels extreme, but the logic applies everywhere. When the prior probability (the probability of having the disease before knowing the test result) is very low, even a very accurate test produces a large number of false alarms. However, if we were to do a second test that is different from the first test but has the same accuracy, then the results are significantly different. If you test positive for both, the chance of you having the disease jumps up to 90%. This shows how much of an impact a second test can have. A second positive result acts as a new piece of evidence that updates the prior probability upward. Bayesian updating is what turns a vague suspicion into a reliable diagnosis.
With this background, we can revisit the debate about Covid testing. During the pandemic, testing was widely accessible and at times required. People took tests before travel, before school, before gatherings, or sometimes simply because they were told it was the responsible thing to do. This meant that enormous numbers of people took tests even when they had no symptoms and no known exposure. In statistical terms this created a situation with a very low prior probability. Most people taking a test at any random moment were not sick.
A low prior means that the value of a positive test becomes weaker. A positive result in someone who has no symptoms and no known exposure is less informative than a positive test in someone who is coughing, feverish, and living with a person who already tested positive. Letting people test with no symptoms will just mathematically lead to many people being falsely tested positive.
One practical limitation also mattered. Many health systems treated a single positive test as a confirmed diagnosis and a second test was almost never required. In some situations this was unavoidable because decisions had to be made quickly. From a statistical perspective, however, a second test would have provided a substantial improvement in accuracy. This is not criticism of emergency policy but simply an observation of the mathematics. A confirmatory test always sharpens the posterior probability whenever the first test is taken in a low prior environment.
Some people argued during the pandemic that we should have assumed a high prior, because the virus spread widely. It is true that large parts of the population were infected at some point, but that does not mean that any random person at any random moment had a fifty percent chance of being infected. Prevalence fluctuated from day to day and from region to region. Using a very high prior when taking a test during a period of low circulation would misrepresent the actual risk. Bayes' rule reminds us that priors are specific to the moment and not to the entire history of the pandemic.
The broader lesson is not that Covid testing was a mistake. Testing played a central role in identifying outbreaks and protecting vulnerable populations. The lesson is that widespread testing without attention to priors has limitations. When a disease is common and spreading rapidly, testing a large group of people is useful. When a disease is more rare in a specific setting, mass testing creates noise rather than clarity. The right strategy depends on the prevalence at the time, the accuracy of the test used, and the purpose of the testing program.
A good Bayesian approach during a pandemic would look like this. People with symptoms or known exposures would be tested first because they have a higher prior probability. If the result is positive, a second test would increase confidence. People without symptoms would be tested strategically rather than constantly, which would reduce the number of confusing or unnecessary positives. Health systems would interpret test results not as simple yes or no answers but as probability updates.
Let us finish on a hopeful note. The same mathematics that helps us understand the limits of testing also gives us reassurance. A single worrying result does not always mean disaster. There is always the possibility that it is a false alarm, especially when the prior probability is low. Bayes' rule offers comfort by reminding us that evidence must be interpreted in context. In everyday life, that is one of the most reassuring pieces of knowledge that statistics can give.