Conditional Probability and Base Rates: Given What, Out of How Many?

Boston's smallpox epidemic of 1721 left a table that OpenIntro Statistics uses to teach conditional probability. Of 6,224 people exposed to the disease, 244 had been inoculated: deliberately exposed to it in a controlled form, which doctors of the time believed could reduce the chance of death. Everything on this page can be read off these six numbers and their totals.

The condition sets the denominator

A conditional probability restricts attention to one group. P(died | inoculated), read "the probability of dying, given inoculation", ignores everyone who was not inoculated. Its denominator is 244, not 6,224.

• P(died | inoculated) = 6 / 244, about 2.5 percent, or 1 in 40.
• P(died | not inoculated) = 844 / 5,980, about 14.1 percent, or 1 in 7.

OpenIntro's definition is P(A | B) = P(A and B) / P(B). Dividing the joint probability (6 of the 6,224 were both inoculated and dead) by the probability of the condition (244 of the 6,224 were inoculated) gives the same 6 / 244. One table yields three kinds of number. A joint probability counts one cell against everyone. A marginal probability counts a row or column total against everyone: 850 of 6,224, about 13.7 percent, died. A conditional probability counts a cell against its own row or column. Demonstrated: it is arithmetic on the table.

The inverse is a different question

Now swap the condition. P(inoculated | died) asks what share of the dead had been inoculated. The answer is 6 of 850, under 1 percent. That answers a different question from 6 of 244. Read carelessly, "fewer than 1 in 100 of the dead were inoculated" turns into "fewer than 1 in 100 inoculated people died". The figure for the inoculated is 1 in 40.

The two directions are tied together by a base rate, here the share of people inoculated at all: 244 of 6,224, about 3.9 percent. Bayes' theorem, which OpenIntro states in the same section, is the formula for turning one direction into the other. The screening-test version of that turn, a rare condition and a test with false alarms, is worked in Judgment Under Uncertainty and Mathematics for Better Decisions. This page stays with the table.

By analogy with logic, mixing up the two directions resembles affirming the consequent, set out in Proof and Precise Reasoning: "if it rained, the road is wet" does not license "the road is wet, so it rained". In the same way, a high P(wet | rain) does not make P(rain | wet) high.

A striking share needs a base rate to beat

Almost every smallpox death, 844 of 850, was uninoculated. That is 99.3 percent, and it sounds like a verdict. Now ask what the share would be if inoculation made no difference. The dead would then look like the whole population, and 96.1 percent of the population was uninoculated. So the striking figure has to be compared with 96.1 percent, not with 50. It does beat it, which fits the lower death rate among the inoculated, but by far less than "99 percent" suggests.

Many headlines have this shape. Take invented numbers: 90 percent of drivers are sober, and drinking multiplies the risk of a crash by five. Sober drivers still make up about 64 percent of crashes: 0.9 against 0.1 × 5 = 0.5, and 0.9 / 1.4 is about 0.64. "Most crashes involve sober drivers" is then true, and it says nothing in drinking's favour. Before you read a share among the cases, find the share in the whole population.

Independence and the multiplication rule

OpenIntro's test for independence is plain: "If two events are independent, then knowing the outcome of one should provide no information about the other." In the table, learning that someone was inoculated moves their chance of dying from 13.7 percent to 2.5 percent, so the two are not independent. Roulette spins are independent: the next spin owes nothing to the last few. Forgetting that is what OpenIntro calls the gambler's fallacy.

The general multiplication rule turns a conditional probability back into a joint one: P(A and B) = P(A | B) × P(B). For the table, 6/244 × 244/6,224 = 6/6,224. A tree diagram draws the same rule. The first branches split by the condition, the second by the outcome, and you multiply along each path. OpenIntro calls trees most useful when processes happen in sequence, each conditioned on the one before.

What the table cannot tell you

A lower death rate among the inoculated does not, on its own, show that inoculation saved lives. OpenIntro points out that the people of Boston chose for themselves whether to be inoculated. The groups may have differed in other ways, and an observational study like this cannot establish a causal connection. The book adds that later research has shown inoculation does reduce death rates. Conditional probabilities describe the table. The page on Cause and Effect asks what it would take for them to measure an effect.

The arts behind it

This is arithmetic carried into probability: a probability is a part over a whole, and the whole is the thing to name. Its typical errors are logical, and the cure is rhetorical: say the denominator aloud. "6 of the 244 inoculated died" is a claim another person can check. "6 died" and "2.5 percent" are not, until you say of whom. As Observe, Define, Represent shows, counts of people are easier to reason with than percentages. In the site's five acts (the site's own synthesis, not a classical scheme), the table trains understanding exactly, checking against evidence and expressing truly.

Where this could be wrong

The strongest objection. The routine taught here, "given what, out of how many, against which base rate?", assumes the right base rate is obvious. It rarely is. All of Boston, everyone exposed, or people of your age and health: each is a defensible comparison group, and each gives a different number.

The best reply. Name the group you used, then check whether the conclusion survives two or three reasonable alternatives. A claim that flips when the group changes was resting on the choice, not on the data.

What would settle it. Nothing in the arithmetic chooses the group, so that choice stays a judgment you must state and defend, not a fact the table supplies. Size is a second limit. Six deaths is a small count, and chance alone could move a rate built on it. OpenIntro's later chapters on inference give the tools for saying by how much.

Try this

1. From the table, compute P(lived | inoculated) and P(inoculated | lived). Write one sentence for each that names its denominator.
2. Find a news sentence of the form "most people who had X were Y". Estimate the share of Y in the whole population, then say whether the claim beats its base rate.
3. Draw the tree: inoculated or not, then lived or died. Multiply along each path and check that the four results add up to 1.
4. Work the probability chapter of OpenIntro Statistics, assigned on the Levels path, and make up one two-way table of your own with a misleading inverse.

Further reading

• David Diez, Mine Çetinkaya-Rundel and Christopher D. Barr, OpenIntro Statistics, 4th ed. (2019), chapter 3, section 3.2, "Conditional probability".
• Judgment Under Uncertainty: Probability, Causation and Forecasting for Bayes' rule in odds form and the psychology of base-rate neglect.
• Systems Thinking: Feedback, Queues, Information and Dynamics for why alarms about rare events are mostly false.