I didn't know, when I wrote this back on August 9, that the bit of Covid innumeracy I was trying to describe has a name: the base rate fallacy. A week earlier, on August 1, Damon Jones, using numbers that make the point perhaps more clearly than I did, had illustrated the fallacy in this brief Twitter thread. I'll summarize for those who don't care to follow the link.

Suppose that 9 out of 10 people who haven't been vaccinated end up in the hospital with Covid. Suppose, further, that there is a vaccine, and that only 1 in 10 of the vaccinated get a case of Covid bad enough to require hospitalization. That's a helluva vaccine, right? Your chance of going to hospital drops from 90 percent all the way down to 10 percent. It's such a good vaccine that 90 percent of the population gets vaccinated. Only 10 percent don't, for whatever reason they find persuasiveâ€”perhaps because they've heard the vaccinated can get sick, too.

Now let's consider the question of who ends up in the hospital with Covid:

Out of every 100 people, 90 will be vaccinated, and, of those, 10 percent, or 9, will end up in the hospital. Ten people will not be vaccinated, and 90 percent of them, or 9, will end up in the hospital. So 18 people in the hospital, half vaccinated, half unvaccinated. It can be terribly misleading to denigrate the efficacy of vaccines by citing the percentage of people in the hospital who've been vaccinated! Don't fall for it. Get vaccinated.

As Jones notes, it's also useful to consider what happens, in the above scenario, if only half the population is vaccinated. Then out of 50 vaccinated people, 5 end up in hospital, and, of the 50 unvaccinated ones, 45 are hospitalized. So now 90 percent of the hospitalized are unvaccinated. There are also 50 people in the hospital, as opposed to just 18.

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