# 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: