Today one of my friends forwarded me this message on Whatsapp:

Israel now provides vaccination status among new coronavirus cases and those who are seriously ill.

New cases (Sat): -

Unvaccinated: 1,006 Partially vaccinated: 39 Fully vaccinated: 1,049

Seriously ill:

Unvaccinated: 71 Partially vaccinated: 7 Fully vaccinated: 133)

Note that I am not sure if the data is reliable here. Let us assume that the given data is accurate. On first glance this can be interpreted to mean that vaccines are inefficient, since the number of vaccinated people among the number of new cases is more than the number of unvaccinated people. And similarly for the number of seriously ill cases. This is dangerous since vaccine skeptics base their arguments on statistics like these.

However, this interpretation is flawed since we disregard one important statistic: the proportion of vaccinated individuals in the population (aka the base rate). If a large proportion of the population has been vaccinated it would imply that the proportion of vaccinated individuals among new cases will also be large. Ignoring the actual proportion of vaccinated individuals is a typical example of the base rate fallacy. This is actually just Bayes theorem in disguise.

To drive this point home we can play around with some numbers. Consider for instance that 85% of Israel’s population is vaccinated and vaccines are 70% effective. Further the probability of a test being positive given that someone is not vaccinated is say 50%. To simplify things we only consider single dose vaccinations. Now we want to find P(vaccinated | cov+) or the probability that someone is vaccinated given that the person has been tested covid positive. Applying Bayes theorem this can be reframed as:

\[P(Vaxed|+)=\frac{P(+|Vaxed)P(Vaxed)}{P(+|Vaxed)P(Vaxed)+P(+|Unvaxed)P(Unvaxed)}\] \[=\frac{0.3 \times 0.85}{0.3 \times 0.85 + 0.5 \times 0.15}\] \[=0.77\]

Thus this shows that given these imaginary but plausible numbers one could expect 77% of covid positive people to have been vaccinated. As more and more people are getting vaccinated we can expect an uptick in the number of vaccinated peoples getting infected, not because vaccines are inefficient but just because so many people are getting vaccinated.