Mathematics 11–12 · Years 11–12

Three cups and a coin: conditional probability and the stick-or-switch game

Probability and data: Conditional probability (Mathematics Advanced, Year 11); Relative frequency and probability (Mathematics Standard 2, Year 12)

Practical, model not builtLow risk

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The idea

When a host who knows where the prize is always reveals an empty cup, switching wins two times in three, which a tree diagram predicts and 60 real trials confirm.

What you need

Three opaque cups and a coin per pair
A die to let the host hide the coin at random (1-2 cup A, 3-4 cup B, 5-6 cup C)
A tally sheet

How to do it

The host hides the coin by rolling the die; the player picks a cup; the host, who knows where the coin is, lifts a different empty cup.
Play 30 rounds in which the player always sticks and 30 in which the player always switches; record wins.
Draw the tree diagram for the first choice and the host's reveal, and compute P(win | stick) and P(win | switch).
Pool the class tallies and compare the relative frequencies with the theory.
Change the rule so the host lifts a random other cup (and a round is void if it shows the coin), play 30 rounds, and explain why switching no longer helps.

What you should see

Sticking wins only when the first pick is right, probability 1/3; switching wins whenever the first pick is wrong, probability 2/3. In 30 rounds the expected wins are 10 when sticking and 20 when switching, each with a binomial standard deviation of 2.6, so one pair's tallies can sit several wins away from 10 and 20 while the pooled class results separate clearly. Under the random-host rule, among rounds that are not void, sticking and switching each win half the time. The learner knows it worked when the pooled switch rate is close to 0.67 and the stick rate close to 0.33.

What changes

What you change: strategy (stick or switch) and the host's rule
What you measure: proportion of rounds won
What you keep the same: coin hidden at random by the die
host always reveals an empty cup
30 rounds per strategy

Common misconceptions

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

Once one cup is lifted the two left are equally likely; the host's knowledge makes the reveal depend on where the coin is.
Switching wins every time; it wins two rounds in three, so losing streaks still happen.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

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 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 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
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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