Mathematics 11–12 · Years 11–12

The sum of two dice: sample space, relative frequency and expected value

Relative frequency and probability (Mathematics Standard 1 and Standard 2, Year 12); Probability and data: Probability (Mathematics Advanced, Year 11); Random variables: Discrete random variables (Mathematics Advanced, Year 12)

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The idea

Thirty-six equally likely outcomes give the sum of two dice a triangular distribution, and relative frequencies from hundreds of real rolls settle towards it as the number of rolls grows.

What you need

Two dice of different colours per pair
A tally sheet and a spreadsheet

How to do it

List the 36 outcomes in a table and count the ways to make each sum from 2 to 12.
Roll the pair 100 times and tally the sums.
Compare the relative frequency of each sum with its probability and compute the expected frequency np for 100 rolls.
Pool the class results and plot the relative frequency of a 7 after 36, 100, 360 and 1000 rolls.
Compute E(X) and Var(X) from the probability distribution and the mean and variance of the pooled data.

What you should see

The ways to make 2 to 12 are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 out of 36, so P(7) = 1/6 and the expected number of 7s in 100 rolls is 16.7 (standard deviation 3.7). E(X) = 7 and Var(X) = 35/6 = 5.83. The relative frequency of a 7 has a standard deviation of 0.062 after 36 rolls, 0.037 after 100, 0.020 after 360 and 0.012 after 1000, so the pooled class value settles towards 0.167. The learner knows it worked when the pooled histogram is triangular and peaks at 7 and the pooled mean is close to 7.

What changes

What you change: number of rolls
What you measure: relative frequency of each sum
What you keep the same: same dice
same rolling method
sums recorded after every roll

Common misconceptions

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

All sums from 2 to 12 are equally likely; there are six ways to make 7 and one way to make 2.
After many rolls the number of 7s must equal exactly one sixth; the relative frequency approaches 1/6 while the count itself can drift further from n/6.

Safety card

Low riskLearners carry it out

Hazards

dice on the floor

Controls

roll inside a tray

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

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 Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Relative frequency and probability; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S1-07
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Relative frequency and probability; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-09
Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Probability and data; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-09
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Random variables; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-07
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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