With all the frustration and horror potential that this pandemic has brought with it for almost two years now, at least those interested in statistics can regularly rejoice: there has rarely been such an abundance of realistic case studies for counterintuitive statistical effects. One that is particularly difficult to swallow can currently be found every week in the RKI's corona report in the table on probable vaccination breakthroughs. According to this, about 45 percent of hospitalized patients were vaccinated in the past month. But wasn't it always said that the unvaccinated would be the ones who would fill the hospitals in the next wave? Does the vaccination actually do anything if it also affects the vaccinated?

Anyone who wants to understand this number has to take into account that in Germany the group of those who have been vaccinated is significantly larger than that of those who have not been vaccinated. But what does that mean? The Israeli psychologists Amos Tversky and Daniel Kahneman came up with a nice experiment back in 1974 to illustrate our intuitive failure in dealing with situations of this kind. They described a fictional character, named Steve, who was a shy, withdrawn, detail-obsessed, and orderly American with no undue interest in the real world. Respondents were asked to estimate whether Steve was more of a farmer or a librarian. Most bet on the latter - forgetting to take into account that the United States has about twenty times as many farmers as librarians.

If you factor in this “a priori probability” of the very different occupational frequencies, contrary to our first intuition, it seems more likely that Steve spends his days on the tractor than between bookshelves.

It is very similar with a hospitalized Covid patient.

The more than six and a half times lower probability of ending up in hospital as a vaccinated person, depending on age, is at least partially offset by the fact that the group of vaccinated adults is currently almost four times as large as that of the unvaccinated.

The inclusion of a priori probabilities in our judgments is therefore essential.

At the same time, we would have preferred to have learned this from more harmless examples than increasingly overloaded hospitals.