The birthday surprise
Fill a room with 5 to 70 people and find the chance that at least two of them share a birthday. See why a class of 23 already has an even chance, and why the number of pairs, not people, is the secret.
Core concept
With n people there are n(n − 1)/2 pairs, each with a 1 in 365 chance of sharing a birthday; the pairs grow so fast that the chance of a shared birthday passes 50% at 23 people: P ≈ 1 − e^(−n(n − 1)/730)
Topic: Maths in everyday life
How many people must be in a room before two of them probably share a birthday? Most people guess about 180, half of 365. The answer is only 23! The secret is to count pairs, not people. With n people there are n(n − 1)/2 pairs, and each pair has a 1 in 365 chance of the same birthday. Ten people make 45 pairs; 23 people make 253. It is easiest to find the chance that no one shares and take it from 1 (the complement). That is close to e^(−pairs/365), so P(a shared birthday) ≈ 1 − e^(−n(n − 1)/730).
What you will learn
- Count the pairs among n people with n(n − 1)/2.
- Use the complement to find the chance of at least one shared birthday.
- Explain why the chance grows much faster than most people expect.
Before you start
- Know that P(an event) = 1 − P(it does not happen).
Materials
- A list of the birthdays in your class (with permission)
Equipment
- None
Procedure
- Record 23 people.
- Record 10 people, then 50 people.
- Record 70 people.
How this simulation is simplified
- 365 equally likely birthdays (no 29 February), and the approximation P ≈ 1 − e^(−n(n − 1)/730).