Mathematics K–6 · Years 5–6

Two dice: why 7 comes up most

Statistics and probability

Practical, model not builtLow risk

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The idea

The sums of two dice are not equally likely because more pairs of faces make 7 than make 2 or 12, and many rolls show the counts settling towards those chances.

What you need

two dice of different colours per pair
a 6 by 6 grid to list every pair of faces
a tally sheet for sums 2 to 12 and a class chart
a spreadsheet with a random-number function (for example RANDBETWEEN(1,6) + RANDBETWEEN(1,6) filled down 3600 rows) for larger trial counts

How to do it

Fill the 6 by 6 grid with the sum for every pair of faces (red 1 with blue 1, red 1 with blue 2 and so on). Count how many cells hold each sum from 2 to 12.
Write each count as a fraction of 36 and predict how many times each sum will appear in 36 rolls.
Roll the two dice 36 times and tally the sums. Compare with the prediction.
Pool the class results (for example 12 pairs, 432 rolls) and compare the pooled tally with the prediction scaled up.
Run 3600 rolls in the spreadsheet and compare the fractions with the grid fractions. Describe how the match changes as the number of rolls grows.

What you should see

The grid holds 1 cell for a sum of 2, rising by one per sum to 6 cells for 7, then falling to 1 cell for 12, so the chance of 7 is 6 out of 36 and the chance of 2 is 1 out of 36. In 36 rolls a pair expects six 7s, and a count from 2 to 10 happens in about 96 percent of runs. Pooled to 432 rolls the expected count for 7 is 72 with a standard deviation of 7.7, so the class total lands between about 57 and 87. At 3600 rolls the fraction of 7s sits within 0.012 of 0.1667 (two standard deviations of 0.0062) about 95 percent of the time, and the tally takes the triangular shape of the grid counts.

What changes

What you change: the number of rolls
What you measure: the fraction of rolls giving each sum
What you keep the same: the same two fair dice
rolled onto a flat surface
every roll recorded

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Every sum from 2 to 12 is equally likely.
A sum of 3 is as likely as a sum of 7 because both need one particular roll.

Safety card

Low riskLearners carry it out

Hazards

dice on the floor

Controls

roll into a shallow tray

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMA3-CHAN-01 MAO-WM-01
Australian Curriculum v9AC9M5P01AC9M5P02AC9M6P01AC9M6P02

Sources

The pages the author read to write this activity.

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