# Conditional probability and imperfect vaccines II

## Despite high coverage, an effective vaccine can still have breakthrough cases.

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On 13th June, a MailOnline headline states:

Study shows 29% of the 42 people who have died after catching the new strain had BOTH vaccinations as cases soar another 40%

In Public Health England’s briefing on 25th June, 50 out of 117 Delta-related (B.1.117.2) deaths had two vaccine doses.

As vaccine coverage rises, the share of Covid-19 deaths with prior vaccinations grows. If everyone had two doses, all people dying (from any cause) would have been fully vaccinated.

Suppose there were 100 people who would die after infection without a vaccine. 95 out of the cohort get vaccinated, and it reduces their likelihood of death by 94%.

The virus attacks. Now: six people die from the disease, whilst the five unvaccinated people also die. That means most people who died from the disease had the vaccine.

Avoid confusing the inverse. These two probabilities are not the same:

• The probability that someone dies with a disease, given that person having a vaccine.
• The probability that someone had a vaccine, given they died with a disease.

With high coverage, immunisation failures can outnumber unvaccinated cases and deaths:

We can see how increasing vaccination coverage affects the vaccinated share. As the vaccine is effective, total deaths from the disease decreases:

Coverage and effectiveness are important statistics for evaluating vaccination programmes. Conditional probabilities can be counter-intuitive.

The code for the graphs is available on R Pubs and GitHub.