Say you read that 57 percent of deaths in a year were people aged 75 or older. It sounds like a warning about old age. It is not a statement about how likely an older person is to die. Those are two different questions, and the gap between them is where a lot of bad statistical reasoning starts.
The Formula Behind the Question
Conditional probability asks how likely one thing is, given that another thing is true. The Stanford Encyclopedia of Philosophy gives the definition: the probability of H given E is the probability of H and E together, divided by the probability of E. This only works when E has a probability above zero.
Bayes' theorem builds on that. It lets you turn around a conditional probability when you know the pieces. In its simple form, the probability of H given E equals the probability of E given H, times the probability of H, divided by the probability of E. The first term is how well the hypothesis predicts the evidence. The second is the base rate, how common the hypothesis is before you look at any evidence. Drop the base rate and you get the wrong answer, however careful the arithmetic.
A Worked Example With Mortality Figures
The Stanford Encyclopedia entry uses U.S. figures for 2000. About 2.4 million of 275 million Americans died that year. That gives an overall mortality rate of 0.00873.
Now narrow the group. About 16.6 million people were 75 or older, and about 1.36 million of them died. Divide 1.36 million by 16.6 million and you get 0.082. So the chance that a senior died that year was about 8.2 percent.
The same answer comes from the formula. The probability of being a senior and dying is 1.36 million over 275 million, or 0.00495. The probability of being a senior is 16.6 million over 275 million, or 0.06036. Divide the first by the second and you land on 0.082 again. The size of the whole population cancels out. What is left is the death rate inside one group.
Turning the Question Around
Now flip it. Of everyone who died, what share were seniors? Take the same joint probability, 0.00495, and divide it by the overall death rate, 0.00873. You get about 0.57. That is the 57 percent from the opening.
Both numbers come from the same data. One says 8.2 percent of seniors died. The other says 57 percent of those who died were seniors. The second is large partly because seniors are a small group with a high death rate, and partly because deaths among the young are rare. Reading 0.57 as a senior's chance of dying overstates the risk by almost seven times.
This mistake has a name. It is called the transposed conditional, and in courtrooms it is often called the prosecutor's fallacy. It swaps the probability of the evidence given a hypothesis for the probability of the hypothesis given the evidence. The Inns of Court College of Advocacy and the Royal Statistical Society have published guidance on statistics for advocates, partly because legal cases lean on figures like these.
Why Counts Beat Percentages
People find these problems easier with counts than with percentages. A Significance magazine piece on screening tests shows this with a Down's syndrome example. About 1 percent of babies are affected. The test is positive 90 percent of the time when a baby is affected, and 1 percent of unaffected babies also test positive. What is the chance a baby is affected, given a positive test?
In the study it describes, only 1 of 21 obstetricians and none of 22 midwives gave the correct answer of about 48 percent. Now picture 10,000 babies. A hundred are affected, and 90 of them test positive. Of the 9,900 unaffected babies, 99 test positive. So 189 babies test positive, and 90 of them are affected. That is 90 out of 189, or about 48 percent. When the same problem was posed in counts like these, 13 of 20 obstetricians got it right, though 7 of 20 still did not.
The lesson is that a positive result from a decent test can still be wrong about half the time, because the unaffected group is so much larger. A related error is treating a test's sensitivity as if it were the chance of disease given a positive result. A paper on positive predictive value notes that physicians often make exactly that slip.
A Short Checklist for Any Statistic
You do not need to run the formula every time you meet a number. You do need to ask a few plain questions first.
- Which group is the denominator? Seniors, all deaths, or everyone?
- What is the base rate of the thing being tested for or described?
- Is the quoted figure the chance of the evidence given the condition, or the chance of the condition given the evidence?
- Can I rewrite it as counts out of 10,000 people?
If the answers are missing, the number is not yet telling you much. The sources behind this piece do not explain how to find a reliable base rate for a real case, which is often the hardest step.
Where the Debate Sits
Bayes' theorem itself is just arithmetic, and nobody disputes the mortality example. The argument is about how far to take it. Bayesians treat degrees of belief as probabilities and update them as evidence arrives. Critics, often from the frequentist tradition, argue that probability should describe long-run frequencies, and that assigning a prior to a one-off hypothesis can be arbitrary. Those are fair objections, and this piece does not settle them. The base-rate lesson holds either way, because the mortality and screening examples use frequencies that anyone can count.
Sources
- Bayes’ Theorem (Stanford Encyclopedia of Philosophy)
- RSS - Statistics law guides
- Statistics and Probability for Advocates | ICCA
- Antibody (Serology) Testing for COVID-19: Information for Patients and Consumers | FDA
- Statistical Guidance on Reporting Results from Studies Evaluating Diagnostic Tests - Guidance for Industry and FDA Staff | FDA
- Diagnostic tests: how to estimate the positive predictive value - PMC
- How to explain screening test outcomes - Significance magazine
- Winton Centre Cambridge
Siobhan O'Reilly investigates the friction between urban decay and digital expansion for Deeperpectives. She spent a decade reporting from transit hubs and temporary settlements across the rust belt. Her work prioritizes the raw data of lived experience over polished corporate narratives.
