by John Allen Paulos
Covid has given rise to a variety of counterintuitive mathematical outcomes. A good example is this recent headline (link below): One third of those hospitalized in Massachusetts are vaccinated. Anti-vaxxers have seized on this and similar such factually accurate headlines to bolster their positions. They, and others as well, interpret them as evidence that the vaccine isn’t that effective or perhaps hardly works at all since even states with very high vaccination rates seem to have many breakthrough infections that lead to hospitalization. Contrary to intuition, however, such truthful headlines actually indicate that the vaccine is very effective. I could cite common cognitive biases, Bayes’ theorem, graphs, tables, and formulas to explain this, but a metaphor involving fruit may be more convincing and more palatable.
Let’s assume we have a very large casserole pan full of cherries and a small bowl of grapes. Say there are 500 cherries in the huge pan and 50 grapes in the small bowl. Further assume that the cherries alone have been treated with a substance that very substantially reduces spoilage and that nevertheless 3 of them are spoiled, as are 7 of the untreated grapes. Now for the perhaps obvious punchline, let’s identify these 550 fruit with people in a highly vaccinated state, the cherries with the vaccinated people, the grapes with the unvaccinated people, and the spoiled 10 pieces of fruit with those who are hospitalized.
Clearly, 30%, a substantial percentage of those hospitalized, had been vaccinated, and 70% of those hospitalized had not been vaccinated. That is, roughly 1/3 of the spoiled fruits (3 of the 10) are cherries, and 2/3 of the spoiled fruit (7 of the 10) are grapes. Just as clearly, this does not show vaccination is ineffective since the chance that a cherry is spoiled is 3/500 or .6%, whereas the chance that a grape is spoiled is 7/50 or 14%. The treatment the cherries received was most effective.
Nevertheless, the headlines that will stick with many people are those like the one at the beginning of this piece truthfully asserting that (almost) 1/3 of those hospitalized had been vaccinated.
The numbers were obviously chosen for ease of computation, but were not cherry-picked to describe an unlikely situation. Quite generally, when most people are vaccinated, we should expect a good fraction of those hospitalized to be vaccinated. In fact, we should look forward to the day when almost all of a very small number of hospitalized people will have been vaccinated.
This might be a difficult point to make, but given the torrent of coverage of vaccinations and hospitalizations, surely a ten-minute segment could be put together with compelling graphics, voiceovers, cherries and grapes to get it across to a large audience. It’s at least as informative as the histogram showing the daily case rate, which is aired over and over again.
John Allen Paulos is a Professor of Mathematics at Temple University and the author of A Mathematician Reads the Newspaper, Innumeracy, and a forthcoming book, tentatively titled Who’s Counting – Puzzles, Probability, and Politics with a Smidgeon of Whimsy.