Two dice and the long run

Years 5–10Stage 3 to Stage 5Run it

Two dice can land in thirty-six equally likely ways but make only eleven totals. Roll them: how often does each total come up?

A six by six grid of the 36 ways two dice can land: the first die gives the row, the second die the column, and each cell shows its total. The 6 cells that make 7 are outlined. Beside it, a column graph of the fraction of rolls for each total from two to twelve is empty: no rolls yet. — The first die picks the row and the second die the column: 36 cells, each an equally likely way for the dice to land.
Key Heavy outlines: the cells that make the total you follow, and its number under the graph. Columns: the fraction of the rolls so far that gave each total, its relative frequency. The darker column is the total you follow. Dashed: each total's share of the thirty-six cells, the probability the grid gives it. Line: the fraction of rolls that gave the total you follow, after each roll; the dot is the latest.

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Roll two dice again and again: which totals come up most, and how many rolls does it take for a pattern to show?

Two dice and the long run

Two dice can land in thirty-six equally likely ways but make only eleven totals. Roll them: how often does each total come up?

The first die picks the row and the second die the column: 36 cells, each an equally likely way for the dice to land.

Key Heavy outlines: the cells that make the total you follow, and its number under the graph. Columns: the fraction of the rolls so far that gave each total, its relative frequency. The darker column is the total you follow. Dashed: each total's share of the thirty-six cells, the probability the grid gives it. Line: the fraction of rolls that gave the total you follow, after each roll; the dot is the latest.

Key Heavy outlines: the cells that make the total you follow, and its number under the graph. Dashed: each total's share of the thirty-six cells, the probability the grid gives it.

What do you think?

Roll the two dice thirty-six times, as many rolls as the grid has cells. How many times will each total come up?
- Exactly as many times as it has cells
- Not exactly: some totals more often, some less often
- Every total about the same number of times
Say why (optional)
Roll ten thousand times and follow the fraction of rolls that are sevens. Where does it end up?
- Near one in eleven, since seven is one of eleven totals
- Near one in six, the share of the grid's cells that make seven
- It keeps swinging up and down and never settles
Say why (optional)
Your last roll was a seven. On the next roll, how likely is another seven?
- Less likely than usual
- Just as likely as usual
- More likely than usual
Say why (optional)

Your explanation

Use the grid and the rolls to explain what decides how often each total comes up, and how the number of rolls changes what you see.

The idea, step by step

2 · Ways to make a totalOne then two is a different cell from two then one, so three has two ways. The dashed outlines show each total's share.
$A six by six grid of the 36 ways two dice can land, each cell showing its total, with the 6 cells that make 7 outlined. Beside it, a column graph of the fraction of rolls for each total from two to twelve, with a dashed outline for each total's share of the grid. After 36 rolls, 7 has come up 4 times, a fraction of 0.111; its share of the grid is 0.167.$
3 · Thirty-six rollsAs many rolls as cells, yet the columns miss the dashed outlines: here five came up more often than seven.
$A six by six grid of the 36 ways two dice can land, each cell showing its total, with the 6 cells that make 7 outlined. Beside it, a column graph of the fraction of rolls for each total from two to twelve, with a dashed outline for each total's share of the grid. After 36 rolls, 7 has come up 2 times, a fraction of 0.056; its share of the grid is 0.167.$
4 · Another thirty-sixA different set of thirty-six rolls gives a different ragged pattern. A small sample wanders; it need not look like the grid.
$A six by six grid of the 36 ways two dice can land, each cell showing its total, with the 6 cells that make 7 outlined. Beside it, a column graph of the fraction of rolls for each total from two to twelve, with a dashed outline for each total's share of the grid. After 10000 rolls, 7 has come up 1621 times, a fraction of 0.162; its share of the grid is 0.167.$
5 · Ten thousand rollsTen thousand rolls: every column sits close to its dashed outline. Sevens are 0.162 of the rolls; the grid gives 0.167.
$A graph of the fraction of rolls that gave 7, after each roll, against the number of rolls from one to ten thousand, each marked number ten times the one before, with a dashed line at its share of the grid, 0.167. After 10000 rolls the fraction is 0.162. Of the 1621 rolls straight after a roll of 7, 274 gave 7 again, a fraction of 0.169.$
6 · Wander, then settleOver the first rolls the fraction of sevens swings widely. As the rolls add up, it settles close to the dashed line.
$A graph of the fraction of rolls that gave 7, after each roll, against the number of rolls from one to ten thousand, each marked number ten times the one before, with a dashed line at its share of the grid, 0.167. After 10000 rolls the fraction is 0.162. Of the 1621 rolls straight after a roll of 7, 274 gave 7 again, a fraction of 0.169.$
7 · No memoryStraight after a seven, the next roll was a seven 0.169 of the time, close to 0.167: the dice do not remember.

Each total: the cells of the grid that make it, its share of the grid, and the rolls so far that gave it

Total	Cells that make it	Share of the grid	Rolls that gave it	Fraction of rolls
2	1	0.028	279	0.028
3	2	0.056	555	0.056
4	3	0.083	853	0.085
5	4	0.111	1099	0.110
6	5	0.139	1403	0.140
7	6	0.167	1621	0.162
8	5	0.139	1404	0.140
9	4	0.111	1127	0.113
10	3	0.083	821	0.082
11	2	0.056	551	0.055
12	1	0.028	287	0.029

Try it in the Lab

Practicals with real materials, each with its safety card.

Two dice: why 7 comes up mostYears 5–6Hands-on mathematicsLow risk
Coin toss: more trials, less variationYears 5–6Hands-on mathematicsLow risk
Rolling one die: relative frequency settles towards one sixthYears 7–8Hands-on mathematicsLow risk
Sum of two dice: 36 equally likely pairs, 11 unequal totalsYear 9Hands-on mathematicsLow risk
Fifty coin tosses: real randomness has long runsYear 9Hands-on mathematicsLow risk

Learn it in a lesson

Assigns probabilities to simple events as fractionsStage 3Find the lesson
Calculates theoretical probabilities for single-step experimentsStage 4Find the lesson
Runs experiments and compares relative frequency with theoretical probabilityStage 4Find the lesson
Designs, runs and evaluates probability simulations using digital toolsStage 5Find the lesson

With a learner

Three questions to ask

How many cells of the grid make a total of seven, and how many make two?
After thirty-six rolls, why do the columns not match the dashed outlines?
You have just rolled a seven. Is another seven less likely next time?

What to expect

Many learners expect every total to be equally likely, or a few dozen rolls to match the grid exactly.

What to try next

Open a practical in Try it in the Lab, above, to roll real dice, tally the totals and compare the class's fractions with the grid.

About this model

What is simplified

The dice are simulated: each roll picks one of the thirty-six cells, every cell exactly equally likely, and each roll is independent of the ones before. Real dice are very nearly fair, but not perfectly.
The rolls come from three fixed sets, A, B and C, so every step, test and printed figure shows the same rolls each time. Choose another set to see a different run of rolls.
Roll the dice makes the first roll straight away, then two a second up to the twelfth, so you can watch each one land: a ring shrinks onto its cell. After that the rolls come faster and faster, up to ten thousand, and the ring is no longer drawn.
The column graph's scale stops at four tenths. In the first few rolls a column can be taller than that: it is drawn to the top of the scale with an arrow above it, and the table gives its value.
The Long run graph spreads the rolls out so that each marked number of rolls is ten times the one before. Spread evenly, the first hundred rolls, where the fraction swings most, would be squeezed against the left edge.

Review

Demonstration: a simplified model. Checked against its written sources, 26 September 2026. Not reviewed by a qualified teacher.

Curriculum references

MA3-CHAN-01MA4-PRO-C-01MA5-PRO-C-01MAO-WM-01AC9M5P01AC9M5P02AC9M6P02AC9M7P02AC9M8P02AC9M8P03AC9M9P01

Reference, not a verified alignment.

All demonstrations Practicals in the Lab