Mathematics 7–10 · Year 9

Fifty coin tosses: real randomness has long runs

Statistics and probability

Practical, model not builtLow risk

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The idea

Genuine random sequences contain longer runs of the same outcome than people expect, so the length of the longest run separates real tosses from invented ones.

What you need

1 coin per learner (any Australian coin)
Recording strip with 50 boxes for H and T
A second strip on which the learner invents a sequence of 50 without tossing

How to do it

Each learner invents a sequence of 50 heads and tails that they think looks random and writes it on strip A.
Each learner then tosses a coin 50 times and records the outcomes on strip B.
Strips are collected, shuffled and redistributed without labels. For each strip the receiver records the longest run of identical outcomes and the number of runs.
Use P(A and B) = P(A) × P(B) to find the probability that the five tosses beginning at a given position all match, (1/2)^4 = 1/16 (the first can be either face and the next four must copy it), then explain why a run of 5 somewhere in 50 tosses is still likely when there are 46 positions where it could begin.
The class sorts strips into likely real and likely invented using the longest-run rule agreed on (a longest run of 5 or more suggests real).
Reveal the labels and count how many strips were classified correctly.

What you should see

For 50 fair tosses the probability that the longest run is at least 5 is 0.821 and at least 6 is 0.544 (computed exactly by counting binary sequences whose runs stay below the threshold); the probability it is at least 4 is 0.981. Invented sequences tend to avoid long runs because people expect a fair coin to alternate, so the class counts how many of its own invented strips reach a run of 5. Using the rule longest run of 5 or more means real, about 82 per cent of real strips are classified correctly, and invented strips are classified correctly whenever their author avoided runs of 5. The learner knows it worked when the real strips' longest runs centre near 6 (the theoretical median, since P(L ≥ 6) = 0.544 and P(L ≥ 7) = 0.309) and the rule separates real from invented strips in the class's own sample.

What changes

What you change: whether the sequence was tossed or invented
What you measure: length of the longest run
What you keep the same: 50 outcomes per strip
same coin type
outcomes recorded in order without skipping

Common misconceptions

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

A random sequence alternates often and never has five heads in a row.
After four heads a tail is more likely.
A long run shows the coin is biased.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

Note

No chemicals or 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 K–10 Syllabus (2022), Stage 5 Probability A; page read 2026-09-22MA5-PRO-C-01
Mathematics K–10 Syllabus (2022), Stage 5 Probability B (Path); page read 2026-09-22MA5-PRO-P-01
Australian Curriculum v9AC9M9P03

Sources

The pages the author read to write this activity.

All Concept Studio activities