Birthday Problem
With 23 people the chance of a shared birthday exceeds 50%.
Key idea
With 23 people the chance of a shared birthday exceeds 50%.
In depth
With only 23 people, the chance two share a birthday exceeds 50%. Pairs grow quadratically: 253 pairs among 23 people. Compute via the complement: P(no match) = (365/365)(364/365)...(343/365).
Worked example
P(no match for 23) ≈ 0.493, so P(match) ≈ 0.507.
Formula
Real-world uses
- Explaining why 23 people in a room often share a birthday
- Checking hash collision risk in computing
Common mistake
Comparing with the chance someone shares your own birthday.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Birthday Problem?
With 23 people the chance of a shared birthday exceeds 50%. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Birthday Problem?
The key formula is P ≈ 1 − e^(−n²/2·365).
Where is Birthday Problem used in real life?
Real-world examples: Explaining why 23 people in a room often share a birthday; Checking hash collision risk in computing.
What should I learn before Birthday Problem?
Make sure you are comfortable with Gambler’s Fallacy, the previous topic in Probability & Statistics.
What is a common mistake with Birthday Problem?
Comparing with the chance someone shares your own birthday.
How can I practise Birthday Problem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Birthday Problem?
Move on to Monty Hall Problem next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.