One Cell, Three Denominators, and Only One Is the Question

You'll learnwhat conditioning does to the sample space, how to read a conditional probability off a row or a column of a two-way table, why P(A given B) and P(B given A) are different questions, how a tree multiplies along a branch and adds across paths, and why a 95% accurate test can be wrong about most of its positives.

Thirty students hold a job and are seniors. That one count answers three different questions depending on what it is divided by: 30 of 200 students (0.15), 30 of 60 seniors (0.50), 30 of 72 job-holders (0.4167). The word "given" picks the denominator, and picking the wrong one is this topic's entire catalogue of errors. Then the same move at scale: a disease test that is positive for 95% of the people who have it, pointed at a population where only 1% do. The 9,900 healthy people send 990 false positives against the sick group's 95 true ones, so a positive result means disease with probability 95/1085 — about 0.088. Nothing is wrong with the test. The condition changed, and the denominator changed with it.

8 STEPS · 6 QUICK CHECKS · THE RESTRICTED WORLD · ROW VS COLUMN · TREE DIAGRAMS · THE BASE RATE · v1

one cell of 30 answers three questions · 95 true positives against 990 false ones · the condition sets the denominator

Before you start

What you're looking at

A two-way table of 200 students — 60 seniors, 72 job-holders, and 30 students who are both. That cell of 30 is about to be divided by three different totals.

The question

"Given that a student is a senior, what is the probability they hold a job?" — and why that is a different question from the one with the two groups swapped.

Watch for

The denominator. Every error in this topic is the right numerator over the wrong total, and the word "given" is what picks the total.

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