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.

8 STEPS · 6 QUICK CHECKS · SAMPLE SPACE · THE TWO RULES · EQUALLY LIKELY · THE COMPLEMENT · v1

36 rolls and 11 sums both total 1 · adding gives 2.0 at ten parts · the complement gives 0.8926

Before you start

What you're looking at

Two complete lists of what two fair dice can do — the 36 ordered rolls, and the 11 possible sums — drawn as two bars of the same length, because both lists total 1.

The question

If both lists are complete and both total 1, why does counting over one of them give the wrong answer?

Watch for

The widths. The 36 pieces are identical; the 11 are not. Totalling 1 is one rule, and being equally likely is a separate claim that has to be earned.

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