Why people aren’t getting vaccinated: Bayes’ Rule misinformation
There have been many significant events that have taken place the past few years with regards to COVID-19. There was the emergence of the disease in the US in the spring of 2020, starting winter of 2021 when the first US residents started getting vaccinated, and the development of the delta variant in the US the past few months. Through all of these unprecedented times, there has been a lot of uncertainty circulating around the virus itself, how it spreads, and the vaccines such as Pfizer, Moderna, and Johnson&Johnson. Despite all of the urgency to get vaccination to “return to normalcy”, there exists a population of individuals who refuse to get vaccinated for numerous reasons, whether it be religious, personal, etc. (in this case, I am not talking about people who are medically exempted from getting the vaccine). One of the biggest arguments for why one shouldn’t get vaccinated goes along the lines of: “the number of people getting COVID who are vaccinated is more than the number who aren’t – this shows the vaccine doesn’t work”. While statement this is technically true – as of October 22nd, 59% of individuals who have COVID currently in Tompkins county are vaccinated – the fundamental concept that these people are missing is that they are looking at the wrong statistics to base this argument off of. This is what Bayes’ Theorem entails; this blog post will dive into what this theorem is and how people are using it incorrectly to base their judgements off. We will use data directly from the Tompkins county government website for COVID and vaccination data.
Bayes’ Theorem is a formula used in probability and statistics to describe conditional events, which is the probability that an event occurs given that another event has already occurred. It can be defined as:
Meaning that the probability of event A occurring given that event B has already occurred equals the probability A and B occur divided by the probability of B occurring. To translate this into COVID data terms, let event C equal probability of someone getting COVID and event V equal the probability that someone is fully vaccinated (post 2-weeks).
The statistics that people should be looking at is the probability of getting COVID given vaccinated: P(C | V)
However, the claim above is the switched version: they are arguing the probability of being vaccinated given having COVID: P(V | C)
If we expand these expressions:
P(C | V) = P(C ∩ V)/P(V)
P(V | C) = P(C ∩ V)/P(C)
We can see that these are very different statistics. Let’s plug in some real data from October 22nd, 2021 in Tompkins county to calculate the exact numbers.
P(V | C) = 0.59
P(V) = 70,622/102,180 = 0.69 (102,180 is the total population in Tompkins county as of 2019). This does not account for Cornell students, so the percentage may be slightly lower in reality. However, the rough estimate is large enough to show the drastic difference in the tow different Bayes’ Theorems above.
P(C) = 6,796/102,180 = 0.067
We can now solve for P(C|V) by first solving for P(C ∩ V).
P(C ∩ V) = P(V | C)*P(C) = 0.59*0.067 = 0.040
Therefore, P(C | V) = P(C ∩ V)/P(V) = 0.04/0.69 = 0.057
There is a 5.7% chance, given the number on 10/22/2021 in Tompkins County, that a person who is vaccinated will get covid. The 59% statistics is simply an incorrect measure; given the fact about 70% of Tompkins County is vaccinated, it probabilistically makes sense that most of those who had COVID on October 22nd are vaccinated. This stresses the importance of a solid statistics foundation to make these life-changing decisions, for yourself and your community.
Source:
https://tompkinscountyny.gov/health/factsheets/coronavirusdata#vaxtotals