Disjoint Is Not the Far Side of Independent. It Is the Far Side of the Line

You'll learnwhat mutually exclusive means in a sample space, why the addition rule subtracts the overlap exactly once, and why two events with positive probability can be disjoint or independent but never both.

Hold two events at 0.50 and 0.40 and let the overlap between them move. Every legal version of the pair is one value of that overlap, and two of them have names. At an overlap of 0 the events are mutually exclusive and knowing B happened drops P(A) from 0.50 to zero. At an overlap of 0.20 — which is 0.50 × 0.40 — they are independent and knowing B happened changes nothing at all. Those are different points on the same line, and the second is the only one where the word "unrelated" applies. The arithmetic half is smaller and just as heavily tested: the overlap you subtract in an "or" question is exactly the overlap disjoint events do not have, and a survey that adds 70% to 55% and reports 1.25 has counted the students who do both of them twice.

8 STEPS · 6 QUICK CHECKS · DISJOINT EVENTS · THE ADDITION RULE · CONDITIONAL PROBABILITY · v1

disjoint at overlap 0 · independent at overlap 0.20 = 0.50 × 0.40 · never both

Before you start

What you're looking at

Two events held at probability 0.50 and 0.40, drawn as two bars on one lane, with the overlap between them free to move from 0 up to 0.40.

The question

Mutually exclusive and independent are both ways of saying two events are "separate". Where does each one actually sit?

Watch for

The union. Every bit of overlap the two bars gain, the extent they cover together loses — which is the addition rule, happening in pixels.

Step 1 / 8