# Conditional probabilities and imperfect vaccines

## Why is it plausible most future Covid-19 hospitalisations are among those vaccinated?

In recent reporting, BBC journalist Marianna Spring showed a sticker left by protesters:

A sticker reads:

“60–70% of hospital admissions and deaths are from people who have had 2 doses of the vaccines.”

That sticker provides a short link to “NHS SOURCE”. The document is not, in fact, from the National Health Service.

It is from the modelling sub-group advising the UK government. Despite the exact quote-marks, those words do not appear in the document. The sticker intends to paraphrase this part:

32. The resurgence in both hospitalisations and deaths is dominated by those that have received two doses of the vaccine, comprising around 60% and 70% of the wave respectively. This can be attributed to the high levels of uptake in the most at-risk age groups, such that immunisation failures account for more serious illness than unvaccinated individuals.

Given the nature of the protest, the implication is vaccines are thus ineffective. **The figures do not refer to current admissions and deaths. Those numbers are about modelling the future.**

**Why is it plausible most hospitalisations are among vaccinated people?**

The answer lies in conditional probabilities and Bayes’ Theorem. **Conditional probabilities** are the chance an event happens, given another event *already* did.

**We should avoid confusing the inverse. **These two probabilities are different:

- The probability of hospital admission, given the person has two vaccine doses.
- The probability a person in hospital with the disease is vaccinated with two doses.

Suppose there were 1,000 people who — if infected — would get a serious illness. Assume a full vaccine course reduces the likelihood of hospital admission by 90%. Here, 950 people (95%) receive that vaccine. The remaining 50 go to hospital without it.

The vaccine is imperfect: among the vaccinated, 95 *also* go to hospital (as the assumed reduction is 90%). **In this calculation, about two-thirds of those hospitalised after infection had vaccinations. **That is different to the probability of hospitalisation after vaccination. That chance is 10% of its non-vaccinated probability.

We can write that paragraph in mathematical notation. Here, ℙ**(V **| **H) **means the probability they had a full vaccine (V), given they are in hospital (H):

This calculation depends on both vaccine coverage and effectiveness.

With wide vaccine coverage, a low proportion of failures can outnumber non-vaccinated people. As the modelling document states:

This is not the result of vaccines being ineffective, merely uptake being so high.

**Probabilities can be confusing and counter-intuitive.**