# The Bayes theorem

## A word about depending probabilities#

Sometimes numbers can lead to a simplification in their interpretation. In week 46 the German DIVI-register published an ICU occupancy in the group of people older than 60 years of 45% in vaccinated patients. That means that at the same time 55% of them have not been vaccinated. That’s not a big difference and so one might think that vaccination will not prevent serious infections.
But there is a dependency that should be considered as well. The vaccination rate in the group of people older than 60 is 86% while only 14% are not vaccinated in this group. Thanks to the British mathematician Thomas Bayes (1701-1761) we can calculate the risk of ICU hospitalization for the two groups.

## Risk assessment for the ICU admission for people older than 60 years#

Let’s put the numbers into a table to give an overview of the data:

Vaccinated Unvaccinated Overall
Vaccination status G = 86% U = 14% 100%
ICU admission in week 46 in % I(g) = 45% I(u) = 55% 100%
Hospital admission in week 46 - - -
Risk to require ICU admission P(g) = ? P(u) = ? P = ?

According to the theorem of Bayes we can calculate the probability (respectively the risk) of an ICU admission for vaccinated people in this age group by:

The same done for the unvaccinated people leads to:

Unfortunately we don’t know the overall risk P, but will see that it is not important for what we want to do. We want to know the ratio of the probabilities of the two groups and therefore we will divide them:

You see that P is no longer important and we can insert our numbers:

Finally we can calculate the factor of the increased risk for an unvaccinated person:

Well, to summarize: the risk of a 60 year or older unvaccinated person is almost 8 times higher as for a vaccinated person. That’s a damn high increase in risk.
Simple mathematics make it clear and I know why I want to get the booster shot.

PS:
DIVI refers to the Deutschen Interdisziplinären Vereinigung für Intensiv- und Notfallmedizin. For none-native speaker it reads terrible but it’s simply the German Interdisciplinary Association for Intensive Care and Emergency Medicine. They are updating the number of hospital and ICU admissions daily.
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