Mathematics 7–10 · Year 10
Monty Hall with three cups: switching wins two times in three
Statistics and probability
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
When the host knows where the prize is and always reveals an empty cup, switching wins whenever the first pick was wrong, which is two times in three, and repeated trials make that visible against intuition.
What you need
- 3 identical opaque cups and 1 coin per pair
- 1 six-sided die per pair to place the coin at random
- Recording sheet: trial, first pick, host reveal, strategy, win or lose
How to do it
- One learner is the host and hides the coin under a cup chosen by a die roll (1 or 2 left, 3 or 4 middle, 5 or 6 right) while the player looks away. The player picks a cup. The host lifts one of the other two cups that is empty (choosing at random when both are empty).
- Run 30 trials where the player always stays and 30 where the player always switches; record wins.
- Pool the class results for each strategy and compute the relative frequency of winning.
- Draw a tree diagram: first pick right (1/3) or wrong (2/3); under switching the player wins exactly when the first pick was wrong.
- Change the rules so the host opens a random cup without knowing where the coin is (and the trial is void if the coin is revealed); run 30 trials and compare.
What you should see
Switching wins with probability 2/3: in 30 switch trials the expected wins are 20 with a standard deviation of 2.58, so 15 to 25 is ordinary; staying wins with probability 1/3, expected 10. Pooled over 10 pairs (300 trials per strategy) switching wins about 200 ± 16 and staying about 100 ± 16, a difference no one attributes to luck. Under the random-host variant the two strategies both win half of the non-void trials. The learner knows it worked when the pooled switch rate lies between 0.61 and 0.72 and the stay rate between 0.28 and 0.39.
What changes
- What you change
- strategy (stay or switch) and host knowledge
- What you measure
- relative frequency of winning
- What you keep the same
- host always reveals an empty cup in the standard version
- coin position chosen by a die roll
- same number of trials per strategy
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- After one cup is removed the two remaining cups are equally likely.
- The host's knowledge makes no difference.
- Thirty trials that come out 17 wins for switching disprove the 2/3.
Safety card
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 B (Path); page read 2026-09-22MA5-PRO-P-01
- Australian Curriculum v9AC9M10P02AC9M10P01
Sources
The pages the author read to write this activity.