# Which percentages are considered rare?

## IESE Professor Sebastian Hafenbrädl on probabilities and percentages in times of Corona

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Mathematical theorem from 1763 more relevant than ever

Risk communication is one of Sebastian Hafenbrädl's research areas. “You have to present statistical information in such a way that political decision-makers and people understand it,” says the professor at the IESE Business School. “People are not particularly good at handling percentages,” explains Hafenbrädl. "If we use absolute numbers instead of percentages, that explains complicated relationships much better".

The discussion about corona tests is an example of this. “Because a test is positive, that doesn't necessarily mean that you are ill. If a test turns out negative, you can still be infected, ”says Sebastian Hafenbrädl. The decisive parameters are sensitivity and specificity. “Sensitivity describes the probability with which a test will recognize a sick person as sick. Specificity tells us how reliably the test recognizes a healthy person as healthy, ”explains Prof. Hafenbrädl. Why is this necessary? "Because tests that are not 100% reliable also test positive people who are not sick at all - and test negative people who are actually very sick". According to IESE Professor Hafenbrädl, reasonable assessments can only be made if both results are taken into account.

“This is the practical application of the famous theorem of the mathematician Bayes from 1763, it is now more relevant than ever”. Bayes' formula makes it possible to calculate the probability with which an event will occur if another related event has already occurred.

This could be explained with a completely different example, namely the extensive controls at airports. "100% sensitivity means you can catch every possible assassin." How? “By pulling out everyone who has any suspicious item on them - that's the belt buckle too. That means I also check people who then turn out to be harmless ”. But to be on the safe side makes a lot of sense, because if a weapon could be smuggled through the security check, it could cost hundreds of lives under certain circumstances. "Personnel and baggage checks at airports are viewed like a test that finds everyone who is sick, but also those who are not sick at all," says Hafenbrädl - 100% sensitivity. "Conversely, the airport control has a low specificity, since the scanners will hopefully find every terrorist with a weapon, but also classify many as suspicious who are not terrorists". That would be as if a corona test did not find everyone healthy.

That is why the discussion about comprehensive corona tests could quickly lead to misunderstandings and confusion, said Hafenbrädl. He gives a numerical example: “Let's assume that such a test would be 100% sensitive, which is rare. If 0.002% of 83 million Germans were infected, the test would find each of these 166,000 infected people. But if we only had one test that was 99% sensitive, then 1660 of the 166,000 infected people would not get a positive test and thus possibly think that they are not infected.

The second important metric of a test is specificity. If the test now has, let us assume, 98% specificity, it recognizes 81.177 million of the remaining 82.834 million people as healthy, i.e. gives them a negative test result for the virus. However, this means that around 1.657 million non-infected people get a positive test result and thus think that they are infected with Corona, even though they are not ”.

IESE professor Hafenbrädl mentions an HIV test as another example, which has a sensitivity of 99.8% and a specificity of 99.9%. If a person from a non-risk group of 10,000 people tests positive, how likely is it that this person is actually infected with the virus? “Most people, including many experienced doctors, would say, of course, that the person is definitely infected. But if it is not, the actual probability is 50% ”.

“This is also easier to explain in absolute numbers, because we can more easily relate them to one another intuitively. In the scientific literature, we therefore call this format of information representation relative frequencies, ”says Sebastian Hafenbrädl. “Let's imagine a sample of 10,000 people from a non-HIV risk group. We know from previous experience with HIV that on average only one of these 10,000 people is actually infected. This is called the prevalence of the virus, which we need to know for the calculation. Because of the sensitivity of 99.8%, this one sick person would also get a positive test, in 99.8% of the tests, ”explains IESE-Prof. Hafenbrädl. And further: "The other 9999 people are healthy, but only 9998 get a negative test result". Thus 1 of the 9999 people tested would get a positive test result. "With 10,000 tests we have two positive results, but only one of them is actually infected with the virus," said Hafenbrädl.

Assistant Professor of Managing People in Organizations
Ph.D. in Management, University of Lausanne, HEC
Sebastian completed his postdoctoral research at the School of Management, Yale University, and at the Faculty of Business and Economics (HEC), University of Lausanne. He received his Ph.D. in Management from HEC Lausanne. He also holds a diploma from the TUM School of Management, Technical University of Munich. His research on judgment and decision making lies at the intersection of psychology and economics. In particular, he focuses on managerial contexts, ethics, and the role played by social and institutional forces.
Areas of interest.
* Judgment and Decision making
* Behavioral Economics
* Organizational Behavior