Conditional Probability: P(A|B) ≠ P(B|A)

Imagine that after the college entrance exam results come out, someone points out that 80% of students admitted to top universities studied more than three hours a day. Soon a slogan is going around the parents’ chat: “Study three hours a day and you have an 80% chance of a top school.” It sounds natural, but it mixes up two completely different numbers.

What conditional probability means

A conditional probability is written P(A|B): “the probability of A, given that B is true.” The B after the bar is what you already know. It narrows your view to just the cases where B holds, and asks how many of those also have A.

Let’s put numbers on it. Suppose a school has 1,000 senior students:

• 100 get into top universities, and 80 of them studied more than three hours a day;
• 400 students in total studied more than three hours a day.

So:

• P(studied hard | admitted) = 80 ÷ 100 = 80%
• P(admitted | studied hard) = 80 ÷ 400 = 20%

Both numbers share the same numerator — the 80 students who studied hard and got in — but the denominators differ: one looks at the 100 admitted students, the other at the 400 hard studiers. “Most admitted students studied hard” does not mean “most hard studiers get admitted.”

Mix-ups in everyday life

You see this reversal everywhere. “Most car accidents happen close to home” doesn’t mean roads near home are more dangerous — we just do most of our driving there. “Many successful founders dropped out of college” doesn’t mean most dropouts become successful founders.

Drag the area chart below to change the size of each event and how much they overlap, and watch P(A|B) and P(B|A) move independently. You’ll see they’re only equal when A and B are equally common to begin with.

Using it in forecasts

Whenever you see “X% of people with Y have Z,” ask yourself: which direction am I actually trying to predict, and what’s the denominator?