Mathematics 11–12 · Years 11–12

Galton board: ten left-or-right bounces, Pascal's triangle and the binomial distribution

The binomial distribution and sampling distribution of the mean: Binomial distributions (Mathematics Extension 1, Year 12); The binomial theorem (Mathematics Extension 1, Year 11); Random variables: Discrete random variables (Mathematics Advanced, Year 12)

Practical, model not builtLow risk

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The idea

A ball that goes left or right with equal chance at each of ten rows lands in bin k in C(10, k) ways out of 1024, so the bins fill in the proportions of Pascal's triangle.

What you need

A Galton board with 10 rows of pegs and 11 bins, and 100 balls or beads
Without a board: 10 coins per learner, each ball simulated by tossing all 10 and counting heads (heads = bounce right)
A tally sheet and a spreadsheet

How to do it

Drop 100 balls one at a time (or record 100 ten-coin tosses) and tally how many land in each bin 0 to 10.
Write row 10 of Pascal's triangle and explain why C(10, k) counts the paths that end in bin k.
Compute P(X = k) = C(10, k) (1/2)^10 and the expected count 100 P(X = k) for each bin.
Compute the mean np and variance np(1 - p) of the model and the mean and variance of your tally.
Pool the class results and compare the relative frequencies with the binomial probabilities.

What you should see

Row 10 of Pascal's triangle is 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1 (total 1024), so the expected counts per 100 balls are 0.1, 1.0, 4.4, 11.7, 20.5, 24.6, 20.5, 11.7, 4.4, 1.0 and 0.1. The distribution has mean 5 and standard deviation 1.58 bins; 65.6 per cent of balls should land in the middle three bins and fewer than 0.2 per cent in either end bin. A single run of 100 balls scatters about these values; the pooled class tally approaches them. The learner knows it worked when the pooled relative frequencies sit close to the binomial probabilities and the middle bin holds about a quarter of the balls.

What changes

What you change: bin number k
What you measure: number of balls in the bin
What you keep the same: same board, level
balls released from the same point
100 balls per run

Common misconceptions

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

Every bin is equally likely because every bounce is 50-50; there are 252 paths to the middle bin and one path to each end.
A ball that has gone right several times is due to go left; each bounce is independent of the ones before.

Safety card

Low riskLearners carry it out

Hazards

small balls or beads on the floor

Controls

collect spilt balls at once
keep small beads away from young children

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 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
Mathematics Extension 1 11–12 Syllabus (2024), Year 11 focus area The binomial theorem; 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 2026-09-22ME1-11-05
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.

All Concept Studio activities