Interactive lab

Dice Lab: watch the long run arrive

Dice are a clean model of randomness. Each roll is independent and fair, which makes them useful for seeing something money decisions often hide: a short run can mislead, and a long run tells you more.

How the odds work

One six-sided die has six faces, and each is equally likely. Two dice make 36 equally likely pairs, but the pairs do not spread evenly across the totals. A total of 7 can be made six ways. A total of 2 or 12 can each be made only one way.

Two six-sided dice: how many of the 36 equally likely pairs make each total
Total	Pairs out of 36	Chance per roll
2	1	2.8%
3	2	5.6%
4	3	8.3%
5	4	11.1%
6	5	13.9%
7	6	16.7%
8	5	13.9%
9	4	11.1%
10	3	8.3%
11	2	5.6%
12	1	2.8%

Read this table as the reference. The lab below tallies real rolls and compares them with these shares.

Roll and tally

Choose how many dice to roll, then roll once, ten times or one hundred times. The tally keeps every roll until you clear it, and each dice count keeps its own tally. Bars show what you have rolled. The short ticks show what fair dice are expected to do.

The Dice Lab needs JavaScript to roll. The table above gives the same odds in full, so the reasoning still works without it.

Set up the roll

Number of dice

No rolls yet. Choose a roll to start the tally.

Tally

Bars show the observed share of rolls for each total. Short tick marks show the expected share for fair dice.

Rolls: 0
Average so far: —
Expected average: —

Rolls by total
Total	Rolls	Observed	Expected

What to notice

Short runs wander. Ten rolls can make a total look common or rare when neither is true. Watch how the bars change between ten rolls and one hundred.
Observed shares move toward the expected shares as rolls accumulate. The gap closes slowly, and patience is the mechanism.
The average total drifts toward the expected average. A single small average is a weather report, not a forecast.

The same reasoning applies to income, spending and returns. Judge a plan across many periods rather than by the most recent one. Dice are a teaching model of randomness only. They say nothing about markets, prices or any decision you face.

For the full reasoning, read the variance section of the field guide.