Before you can compute a single probability you have to answer a duller question: what could possibly happen? Write down every outcome of the experiment and you have the sample space. Circle the outcomes you care about and you have an event. When — and only when — those outcomes are equally likely, the probability is just a count divided by a count. Everything below is enumerated and counted live, and then actually rolled a few thousand times so you can watch the arithmetic come true.
The sample space
Grouped by the total
outcomes in the eventoutcomes not in itobserved in the rolls
The long run
the counted probabilityrunning proportionordinary spread of chance
The sample spacepick an experiment
6
The event — which outcomes count as a success
7
Outcomes in all—
In the event—
Probability—
Observed so far—
After
Happened
Observed
Counted
Gap in pts
Gap in count
Read the last two columns against each other. As the trials pile up the percentage gap is squeezed toward zero, while the raw count gap usually drifts further from zero. Both of those are what the long run actually promises, and only the first of them is what people think it promises.
The whole of this section rests on one sentence: a probability is a count of the outcomes you want divided by a count of the outcomes there are — and it is only true when every outcome in that list is equally likely. So the real work is never the division. It is building the list. Two habits make the list come out right. First, keep the outcomes distinguishable. Rolling a 2 and a 5 is not one outcome, it is two — (2,5) and (5,2) — because the dice are separate objects even when they are the same colour. Every cell in the grid beside you is one such pair, which is why there are 36 of them and not 21. Second, count by multiplying. Two dice give 6 × 6; three coins give 2 × 2 × 2; a die and a coin give 6 × 2. Each independent stage multiplies the size of the space rather than adding to it, which is why the coin space doubles with every extra coin and gets away from you so fast — ten coins is already 1,024 outcomes. Slide the coin count up and watch the tiles multiply. That doubling is the reason we eventually stop enumerating and start using formulas, but the formulas are only ever shortcuts for this grid.
Five ways this goes wrong. First, and by far the most common: counting the summaries instead of the outcomes. Two dice can total anything from 2 to 12, which is eleven possibilities — so surely the probability of a total of 7 is 1/11? No. The eleven totals are not equally likely; the 36 pairs are. Seven happens six different ways and twelve happens exactly one way, so 7 is six times as likely as 12. The Grouped by the total chart stacks them up and the mistake becomes visible at a glance. Second, assuming equally likely without earning it. The counting formula is a statement about a fair die and a fair coin. Nothing in the arithmetic can tell you the die is fair — that is a claim about the physical object, and if it is false every number on this page is false with it. Third, the gambler’s fallacy. The dice have no memory. After a run of five sixes the chance of another six is exactly what it always was; the running line settles down not because bad luck gets repaid but because early results get diluted by the sheer weight of later ones. Fourth, expecting the count to even out. It does not, and it is not supposed to — watch the last column of the table wander away from zero while the column beside it closes in. Fifth, reading a long-run proportion as a promise about the next trial. “About one roll in six” describes thousands of rolls. It says nothing whatever about the one you are holding in your hand.