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 with corona ending up in the intensive care unit is more than 10 times greater for those who have not been vaccinated than for those who have been vaccinated.