Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

The birthday problem, extended: what about birthdays within a day?

AA SLAA HLAI SLAI HL Solid Also in: Pure maths, Statistics

Research question

How many people are needed for a 50% chance that two share a birthday within one day of each other, and how does using real (non-uniform) birth frequencies change the classic answer of 23?

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Why it makes a good exploration

The basic birthday paradox is too simple for an IA, but its extensions are genuinely hard and interesting: near-matches need careful counting, and real births are not uniform across the year.

The mathematics you'll need

Complementary probability and products
Counting arrangements for near-matches (circular year)
Simulation to check exact results
Using real birth-frequency data

Where the data comes from

Use national births by day or month (ONS publishes UK births by date) and test against class or team birthday lists.

Office for National Statistics (UK) — UK population, earnings, inflation, births and deaths time series.

Real data for this idea

Primary source: US births per day (NCHS and SSA, via FiveThirtyEight): US births by day, 1994–2014. Two CSVs (1994–2003 from NCHS, 2000–2014 from SSA) give year, month, day of month, day of week and births. Licence: CC BY 4.0 (FiveThirtyEight's data repository); cite NCHS and SSA as the original sources.
Also useful: Office for National Statistics (UK): Births in England and Wales. The release tables include births by month of occurrence. Licence: Open Government Licence v3.0: free to reuse with an acknowledgement.
Optional tool: RareBirthday. Browse how common each birthday date is; cite the NCHS data, not the tool. Explore only; cite the NCHS data, not the tool.

If you query CDC WONDER for more recent US births yourself, its data-use rules forbid publishing any count of 9 or fewer, so merge small cells before you show them. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Reproduce the classic result of 23 with explanation.
Derive or approximate the near-match probability.
Simulate to check.
Recompute with real birth frequencies.
Reflect on twins, leap years and school cohorts.

Pitfalls that cost marks

Only doing the classic problem.
Ignoring the circular nature of the year for near-matches.
Simulations with too few trials.

Showing personal engagement

Test predictions on real groups (sports teams' birthdays).
Explain why intuition fails so badly here.
Ask the question for your own school's size.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Complementary probability and products; Counting arrangements for near-matches (circular year)
AA HL	Good fit	Complementary probability and products; Counting arrangements for near-matches (circular year)
AI SL	Good fit	Complementary probability and products; Counting arrangements for near-matches (circular year)
AI HL	Good fit	Complementary probability and products; Counting arrangements for near-matches (circular year)

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: only doing the classic problem; and ignoring the circular nature of the year for near-matches. Say how each one affects your answer.

Use of mathematics (E)

SL: complementary probability and products; counting arrangements for near-matches (circular year); simulation to check exact results; using real birth-frequency data, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Find the expected number of people needed before the first match, and compare with simulation.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL probability exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Complementary probability and products, counting arrangements for near-matches (circular year), simulation to check exact results and using real birth-frequency data.

Where can I get data for this IA?

Use national births by day or month (ONS publishes UK births by date) and test against class or team birthday lists. Checked sources and tools: US births per day (NCHS and SSA, via FiveThirtyEight) and Office for National Statistics (UK).

Turn this idea into your IA

The planner opens with “The birthday problem, extended: what about birthdays within a day?” filled in: the question, the maths and the data sources.

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