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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

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

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).