Independence Is a Passed Test, and the Shortcuts It Licenses

You'll learnthe three equivalent tests for independence and why any one settles it, what the general multiplication rule always says and what independence lets it shorten to, why replacement is what decides whether draws are independent and what the 10% condition buys, and how unions and at-least-one questions combine the addition rule with the complement.

Of 400 adults, 240 subscribe to a streaming service and 220 own a smart TV; 132 do both. Related purchases, surely — yet 132 of 240 subscribers own one (0.55) and so do 220 of 400 adults overall (0.55). Conditioning changed nothing, and that is the definition: the events are independent, however connected they sound. The test decides, not the story, and everything downstream is a license that test grants — multiplying two plain probabilities, treating four draws from a city of 60,000 as unrelated, computing "at least one" as one minus none. Take the shortcut without the license and the arithmetic gives no warning at all: it just returns 0.16 where the answer is 0.1333, or 0.90 where the answer is 0.70. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE THREE TESTS · THE UNIT SQUARE · THE 10% CONDITION · UNIONS AND AT LEAST ONE · v2

Three tests on one table, and a split line that breaks when the world does

Before you start

What you're looking at

A two-way table of 400 adults — 240 streaming subscribers, 220 smart-TV owners, 132 both — and later a square whose width and height are two draws.

The question

Does knowing one event happened change the probability of the other, and which shortcuts are legitimate exactly when the answer is no?

Step 1 / 12