*That 95 percent efficacy number is not the whole story鈥�. In real life, the threat to the vaccinated will depend on the overall prevalence of COVID-19

Cathy O’Neil

I have a friend who works in the New York City Department of Education鈥檚 COVID-19 鈥渟ituation room鈥� 鈥� tracing cases, informing contacts and so on.

She鈥檚 really good at her job, which is why I was surprised to hear her make a strange statistical assertion: Since the Pfizer and Moderna vaccines are 95 percent effective, one in 20 vaccinated people are going to get COVID.

That鈥檚 wrong, dangerously so. And given that an expert who works in the area every day can make such a mistake, I figure it鈥檚 worth some explaining.

The intuition is pretty straightforward. You can get COVID only if you鈥檙e exposed. At the current low level of cases in New York, and given reasonable precautions, you would expect only a small fraction of the population to be exposed.

Hence, if those people are also protected by a vaccine, the actual infection rate should be much lower than one in 20.

How much lower? Let鈥檚 consider the clinical trial that measured the Pfizer vaccine鈥檚 efficacy. There were two groups of about 22,000 people each.

They were similar in all ways 鈥� age, health, location, activities 鈥� except that one group received a real vaccine and the other received a fake one.

Over time, 162 unvaccinated participants developed at least one symptom and tested positive, while only 8 people in the vaccinated group did.

That鈥檚 95% fewer infections, suggesting 95% efficacy. But less than 1 in 2,700 of the vaccinated participants fell ill.

In real life, the threat to the vaccinated will depend on the overall prevalence of Covid. In early January (2021), when an estimated one in 34 people in the US were infected, one could have expected inoculated people to have about a one in 700 chance of contracting Covid.

Now that cases are down to less than one in 100, vaccinated people are even safer.

Keep in mind, too, that so far we鈥檝e been talking only about infections. Vaccinated people are even less likely to actually die of Covid.

And that鈥檚 where the evidence is extremely good. According to the government website tracking 鈥渂reakthrough cases鈥� in the general population, only 88 out of 87 million fully vaccinated people in the US had been reported dead as of April 20, and 11 of those were unrelated to COVID. That鈥檚 a death rate of less than one in a million.

Of course, there are caveats. In the wild, the vaccinated aren鈥檛 necessarily comparable to the unvaccinated. They might be more careful types generally, which would contribute to their lower infection rates.

Or they might take greater risks on the assumption that they鈥檙e protected. Deaths might be undercounted, and in any case will increase over time as more fully vaccinated people get exposed.

And even an excellent vaccine can鈥檛 prevent cases like the one in Kentucky, where a nursing home worker refused inoculation and caused a deadly outbreak.

Still, the COVID-19 Vaccines are a miracle of science. It鈥檚 enough to make me want to require vaccines next school year for all eligible students and staff, even if it puts my good friend out of a job.

Cathy O鈥橬eil is a Bloomberg Opinion columnist and mathematician who has worked as a professor, hedge-fund analyst and data scientist, and author of 鈥淲eapons of Math Destruction鈥�.