Two Lists That Both Total One, and Only One of Them Is Fair

You'll learnwhat a sample space is and how to count over the right one, why the two rules every probability obeys are working tools rather than decoration, when equally likely may be assumed, and why every "at least one" question goes through the complement.

Roll two fair dice. The 36 ordered rolls are a complete list of what can happen and they total 1. The eleven possible sums are also a complete list, and they also total 1 — and a sum of 7 arrives six ways while a sum of 2 arrives one way, so the second list is not equally likely and counting over it gives the wrong answer while passing the check most students know how to run. Then the other half: four parts each failing with probability 0.2, where adding gives 0.8 and the complement gives 0.5904. The difference is not a rounding error. It is exactly the expected number of failures beyond the first, and at ten parts the same addition returns 2.0 — a perfectly good number that is not a probability at all. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · SAMPLE SPACE · THE TWO RULES · EQUALLY LIKELY · THE COMPLEMENT · v2

Both lists total 1, and only the 36 rolls are equally likely

Before you start

What you're looking at

Every way two fair dice can land, drawn as a table of 36 cells, and later as bars whose full length is a total of 1.

The question

The 36 rolls and the 11 sums are both complete lists and both total 1, so why does counting over one of them give the wrong answer?

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