Why is it that we now (mid summer 2021) see high percentage of positive cases in vaccinated population? Shouldn't we see the opposite? Is this the proof that vaccines don't work? Not at all. The reason? Math!
I borrowed the title from Dr Linda Dykes' PDF. And the math below from an article on Czech media DeníkN. The principle described in both cases is can be verified using elementary school math.
The math
In a highly vaccinated country, it is statistically inevitable that the vast majority of new cases will be vaccinated. The effectiveness of vaccines does not call that into question. Let's say that our model population has 100 people. From those 66 is vaccinated and 34 not.
Let's also say that the number of sick people is the same: 10 in vaccinated population and 10 in unvaccinated. Which means 15% of vaccinated and 30% of unvaccinated (all these figures so far are proportional to the current real life situation in Israel, we did not made any numbers up).
Which means that half of the sick people are vaccinated, but if you do the math (
(1 - 10/66) * 100 = 84.8484) the efficacy of the vaccine is still ~85% as promised.This optical illusion will be repeated in more countries and will always provoke a debate about the meaning of vaccination, so it's good to remember that. This is an important explanation for people who feel that vaccination is not working.
Rough translation from DeníkN
Other sources
Since this topic is getting more popular as the countries are increasing the vaccination rate, more well-founded places write about this topic. If you want to google for this phenomena, I recommend using the term "base rate fallacy + covid19 vaccine".