Birthday Problem
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Tap a term to see what it means, or .
Calculate
Example: in a room with 23 people, what is the chance that at least two share a birthday? Intuition says it is small, since the year has 365 days. Change the values to work out your own case.
Step by step
What each term means
- Probability of a Match (P(A))
- The chance that at least two of the n people share a birthday. It assumes birthdays are independent and every day is equally likely.
- Complement (1 −)
- Counting matches directly would mean adding up cases with one pair, two pairs, a triple and so on. It is simpler to compute the opposite, all birthdays different, and subtract it from 1.
- Factorial of the Days (d!)
- Together with (d − n)!, it counts the ways to give each person a different day: d choices for the first, d − 1 for the second, down to d − n + 1 for the last. With d = 365, the year without February 29.
- Days Left Over ((d − n)!)
- Cancels the factors of d! that go unused: of the d days, only n are taken. The fraction d!/(d − n)! is the product d · (d − 1) · … · (d − n + 1).
- Total Possibilities (dⁿ)
- Every way to assign birthdays to the n people, with repeats allowed: each one has d possible days. The whole fraction is the probability that all days are different.