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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE RESTRICTED WORLD · ROW VS COLUMN · TREE DIAGRAMS · THE BASE RATE · v2

One cell of 30 answers three questions; the condition sets the total

Before you start

What you're looking at

A two-way table of 200 students — 60 seniors, 72 job-holders, and 30 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 is that a different question with the two groups swapped?

Step 1 / 12