In the RKI weekly report of 11.11.2021 it is listed on p. 22 that \(36\%\) the over 60-year-old corona patients in the intensive care unit were already fully vaccinated. In this age group, \(87\%\) completely vaccinated at this point in time (see p. 18).
May be:
- \(G\): Over 60-year-olds are vaccinated
- \(U\): Over 60-year-olds are not vaccinated
- \(I\): Over 60-year-olds are in intensive care
Now is
$$P(G) = 0,87 \wedge P(U) = 0,13.$$
Also is
$$P(G|I) = \frac{P(G \cap I)}{P(I)} = 0,36 \wedge P(U|I) = \frac{P(U \cap I)}{P(I)} = 0,64.$$
So is
$$P(G \cap I) = 0,36 \cdot P(I) \wedge P(U \cap I) = 0,64 \cdot P(I)$$
and because of
$$P(I|U) = \frac{P(I \cap U)}{P(U)} = \frac{P(U \cap I)}{P(U)} = \frac{0,64 \cdot P(I)}{0,13} \Rightarrow P(I) = \frac{0,13 \cdot P(I|U)}{0,64}.$$
It follows
$$P(I|G) = \frac{P(I \cap G)}{P(G)} = \frac{P(G \cap I)}{P(G)} = \frac{0,36 \cdot P(I)}{0,87} = \frac{0,36 \cdot \frac{0,13 \cdot P(I|U)}{0,64}}{0,87} = \frac{0,36 \cdot 0,13}{0,64 \cdot 0,87} \cdot P(I|U) \approx 0,08 \cdot P(I|U).$$
This means that the risk of people over 60 ending up in intensive care with Corona is more than 10 times higher for unvaccinated people than for vaccinated people.