Mathematics 11–12 · Year 11

Shared birthdays in a class: counting with permutations

Permutations and combinations (Mathematics Extension 1, Year 11); Probability and data: Probability (Mathematics Advanced, Year 11)

Practical, model not builtLow risk

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

The chance that nobody in a group shares a birthday is an ordered selection of distinct days divided by all possible selections, and it falls below one half at only 23 people.

What you need

Each learner's birthday as day and month only, written anonymously (anyone may use a day drawn from a random number generator instead)
A calculator with nPr, a spreadsheet

How to do it

Before collecting data, each learner guesses the chance that two people in the class share a birthday.
Collect the anonymous day-and-month list and check for a shared day.
Assuming 365 equally likely days, write P(no shared birthday) = 365Pn / 365^n and P(shared) = 1 - 365Pn / 365^n.
Evaluate P(shared) for n = 10, 23, 30 and 50 and find the smallest n for which it exceeds one half.
Pool the results of several classes (or run the model many times) and compare the proportion of classes with a shared birthday with the formula.

What you should see

P(shared) is 0.117 for 10 people, 0.507 for 23, 0.706 for 30 and 0.970 for 50; 23 is the smallest group for which a shared birthday is more likely than not. In many classes of 30 about 7 in 10 contain a shared birthday. The assumption of 365 equally likely days ignores 29 February and seasonal patterns in births. The learner knows it worked when the class result is compared with 0.706 as a probability, not as a guarantee.

What changes

What you change: the group size n
What you measure: the probability of at least one shared birthday, and the proportion of groups in which a day is shared
What you keep the same: day and month only, one entry per person, collected anonymously
365 equally likely days assumed for every n
each guess recorded before the data are seen

Common misconceptions

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

A class needs about 183 people for an even chance, half of 365; any pair can match, and 30 people make 435 pairs.
The chance that someone shares your birthday is the same as the chance that any two people share one; the first is much smaller.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

Note

No chemicals and no heat: the NSW Department of Education Chemical Safety in Schools package does not apply. Collect day and month only, without names or years, so no learner's date of birth is recorded.

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 11 focus area Permutations and combinations; 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 2026-09-22ME1-11-04
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
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities