Birthday Collisions and the Closest Pair of Birthdays

Assume birthdays are independent and uniformly distributed through a 365-day year, with leap day ignored. ### Clarifying Questions to Ask - Is a birthday represented as one of 365 discrete dates or as a continuous time within the year? - In the “gap” question, do we want the minimum gap between any pair, the average adjacent gap, or a guaranteed upper bound? - Should people born on the same date count as a gap of zero? ### Part 1 — Collision probability For (n) people, derive the probability that at least two share a birthday. #### What This Part Should Cover - Use the complement event that every birthday is distinct. - State the valid range of the product and handle (n>365). ### Part 2 — The fifty-percent threshold Estimate the smallest (n) for which the collision probability is at least (1/2). Show both an approximation and the familiar exact threshold. #### What This Part Should Cover - Apply a controlled approximation to the no-collision product. - Translate the approximation into a numerical value and check the adjacent integers. ### Part 3 — Closest birthdays among 300 people Explain why “the gap between any two birthdays” is ambiguous. Under a continuous uniform model, derive the expected minimum gap between adjacent ordered birthdays among (n=300) people. Contrast that with the discrete-day model. #### What This Part Should Cover - Distinguish minimum spacing from the average spacing of roughly (365/n). - Give the continuous expected-minimum-spacing scale and interpret it in time units. - Explain what same-day collisions imply under the discrete model. ```hint Use the complement twice For collisions, calculate the probability of all distinct dates. For minimum continuous spacing, calculate the probability that every internal spacing exceeds a small value and integrate that survival probability. ``` ### What a Strong Answer Covers - Correct probability formulas and approximations. - Explicit modeling choices rather than silently switching between discrete and continuous birthdays. - A defensible explanation for a closest-pair gap much smaller than one day. ### Follow-up Questions - How does a nonuniform seasonal birthday distribution change the collision probability? - What is the probability of at least one shared birthday among 300 people? - How would you estimate the entire distribution, not just the expectation, of the closest continuous gap?