What is going on?
The surprise comes from counting every pair, rather than comparing everyone with one particular person. A room of 23 people contains 253 different pairs. Many small chances can add up to a large chance of at least one match.
Work through it
- Start with the easier event: nobody shares a birthday.
- The first person can have any birthday. The second must avoid one day; the third must avoid two. Multiply these chances.
- Subtract that product from 1 to find the chance of at least one shared birthday.
Put numbers to the idea
For 23 people, the probability of all birthdays being different is about 0.4927. The chance of at least one match is therefore 1 − 0.4927 = 0.5073, or 50.73%. At 50 people, it is about 97.04%.
A little more clarity
Is the birthday paradox really a contradiction?
No. It is an exact probability result that often conflicts with our first guess. The word paradox describes the surprise.
How many people guarantee a match?
With 365 possible birthdays, 366 people guarantee at least one match. Below that number, a match can be very likely without being guaranteed.
Why is matching my birthday less likely?
That question involves only n − 1 comparisons. Any shared birthday involves n(n − 1)/2 pairs, a much larger set.