^{ Type post AuthorGary Smith Date September 20, 2021 Categorized Mathematics, Medicine and Health Tagged Bayes' Rule, COVID-19, COVID-19 vaccines, inverse probabilities, Simpson's Paradox, Thomas Bayes }

## COVID-19, Bayes’ Rule, and Simpson’s paradox

_{Israeli data, when studied carefully, confirm the effectiveness of COVID-19 vaccines}

_{ Gary Smith September 20, 2021 Mathematics, Medicine and Health }

Israel has a very high COVID-19 vaccination rate and yet, on August 15, 2021, 58% of those Israelis hospitalized for COVID-19 were fully vaccinated — suggesting that vaccinations are ineffective or even harmful. This is a great example of two common statistical traps. The first is confusion about inverse probabilities. One hundred doctors were once asked this hypothetical question: In a routine examination, you find a lump in a female patient’s breast. In your experience, only 1 out of 100 such lumps turn out to be malignant, but, to be safe, you order a mammogram X-ray. If the lump is malignant, there is a 0.80 probability that the mammogram will identify it as malignant; if the lump is benign, there is a 0.90 probability that…