Mathematics 11–12 · Year 12

Dice decay: exponential decay and half-life with 100 dice

Applications of calculus: Rates of change (Mathematics Advanced, Year 12); Algebraic relationships: Exponential relationships (Mathematics Standard 2, Year 12); The binomial distribution and sampling distribution of the mean: Binomial distributions (Mathematics Extension 1, Year 12)

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The idea

When every item has the same chance of being removed each round, the number left falls by a constant fraction per round and halves after a fixed number of rounds.

What you need

100 six-sided dice (or 100 coins for the comparison run)
A tray with a lip, a recording sheet, graph paper or a spreadsheet

How to do it

Throw all 100 dice, remove every die showing a 6 and record how many remain (throw 1).
Throw the remaining dice, remove the 6s again, and repeat until fewer than 5 remain, recording after every throw.
Pool the results from all groups (scaled to 100 dice) and plot dice remaining against throw number.
Read the half-life in throws from 100 to 50 and from 50 to 25 and check the two readings agree.
Fit N = 100 (5/6)^n, rewrite it as N = 100 e^(kn), find k, and compare the predicted half-life with the graph.
Repeat with coins, removing heads, and compare the half-life.

What you should see

Expected dice remaining are 83.3, 69.4, 48.2 and 23.3 after 1, 2, 4 and 8 throws; k = ln(5/6) = -0.1823 per throw and the half-life is ln 0.5 / ln(5/6) = 3.80 throws, so the curve crosses 50 between throws 3 and 4 and 25 between throws 7 and 8. Coins halve every throw. The number left after a throw is binomial, so a single group of 100 dice scatters about the expected curve with a standard deviation of 3.7 dice after one throw and 5.0 after four; pooling six groups (scaled to 100 dice) cuts that to 1.5 dice after one throw and 2.0 after four. The learner knows it worked when successive half-lives read the same number of throws.

What changes

What you change: throw number
What you measure: dice remaining
What you keep the same: removal rule (6 only)
all dice thrown together
starting count 100

Common misconceptions

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

The same number of dice is removed every throw; the number removed is proportional to the number left.
After two half-lives nothing is left; a quarter remains.
One run should match the curve exactly; a single run wanders about the expected curve and only pooling smooths it.

Safety card

Low riskLearners carry it out

Hazards

dice on the floor

Controls

throw into a lipped 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 Advanced 11–12 Syllabus (2024), Year 12 focus area Applications of calculus; 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-12-06
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Algebraic relationships; 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-S2-01
Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area The binomial distribution and sampling distribution of the mean; 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 2026-09-22ME1-12-06
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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