I had a little argue recently with one anti-vaxxer and I thought it will make an interesting blog content. I tried to convince them that vaccinating is good because people are at least not dieing and what I’ve heard was:

Ohh, vaccines are not helping at all. A friend of mine is working at the hospital at says that 50% of covid patients were already vaccinated.

I decided not to argue nor check the data source, but rather give it a try. What is important, it took place in the UK where over 70% of population received at least one dose of vaccine.

Let’s say that we have a population of $1\ 000\ 000$, from which $700\ 000$ is vaccinated and $300\ 000$ is not. Now suppose that out of this million, $1\ 000$ people ended up in hospital, $500$ vaccinated and $500$ not. So the probability of ending up in hospital while being vaccinated is $p_v = \frac{500}{700\ 000} = \frac{5}{7\ 000}$ and probability of ending up in the hospital while not being vaccinated is $p_{nv}= \frac{500}{300\ 000} = \frac{5}{3\ 000}$. We can see that the second probability is $\frac{p_{nv}}{p_{v}} = \frac{7}{3} \approx 2.334$ over twice as big!

This clearly shows the positive effect of vaccines even though half of people taken to the hospital has been vaccinated. To proof vaccines being non-effective 70% of people taken to the hospital because of covid should be vaccinated. This might be confusing at first, but makes since after you think for a while.

Calculating above probabilities and understanding the underlying phenomena is closely related to the Bayes’ theorem. Probably the best online resource to understand the topic is a beautiful video from 3blue1brown: