The birthday paradox is the surprising result that a group of only 23 people has about a 50% chance of containing at least one shared birthday. It is called a paradox because most people expect the required group size to be much larger. The key idea is that we are not comparing everyone to one chosen person, but comparing every pair of people in the room.
As the number of people grows, the number of pairs grows quickly, making matches much more likely.
Key Facts
- Number of pairs in a group of n people: C(n, 2) = n(n - 1)/2
- Probability of at least one shared birthday = 1 - P(no shared birthdays)
- P(no shared birthdays among n people) = (365/365)(364/365)(363/365)...((365 - n + 1)/365)
- For n = 23, P(no shared birthdays) is about 0.493, so P(at least one match) is about 0.507
- The 50% threshold occurs at 23 people, assuming 365 equally likely birthdays and ignoring leap years.
- The chance of at least one match rises rapidly because the number of possible pairs grows quadratically with group size.
Vocabulary
- Birthday paradox
- The surprising probability result that a relatively small group can have a high chance of at least two people sharing a birthday.
- Complement
- The complement of an event is the outcome that the event does not happen, and its probability is 1 minus the event's probability.
- Pair
- A pair is a group of two people whose birthdays can be compared for a match.
- Combination
- A combination counts selections where order does not matter, such as choosing 2 people from a group.
- Independent assumption
- The independent assumption treats each person's birthday as not affecting anyone else's birthday.
Common Mistakes to Avoid
- Comparing everyone only to one specific person, which is wrong because the birthday paradox counts matches between any two people in the group.
- Adding probabilities as if each comparison were independent, which is wrong because birthday comparisons in the same group overlap and affect each other.
- Forgetting to use the complement, which makes the calculation harder because it is simpler to find the chance that all birthdays are different first.
- Assuming 23 people guarantees a shared birthday, which is wrong because 23 people only gives about a 50% chance, not certainty.
Practice Questions
- 1 How many unique pairs of people are there in a group of 23 people? Use C(n, 2) = n(n - 1)/2.
- 2 For a group of 10 people, write the exact expression for the probability that all birthdays are different, assuming 365 equally likely birthdays and no leap years.
- 3 Explain why the birthday paradox becomes likely much faster than many people expect, even though there are 365 possible birthdays.