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Birthday Collisions and the Closest Pair of Birthdays

Virtu FinancialQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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This problem tests your understanding of probability theory, specifically the generalized birthday paradox, and your ability to analyze spacing distributions of random variables.

Part 1: The Threshold

Consider a calendar year of N = 365 days. Let P(n, N) be the probability that at least two people in a group of size 'n' share the same birthday. Derive the condition to determine the minimum 'n' such that P(n, N) > 0.5. Using the Taylor expansion e^(-x) ≈ 1 - x, provide an approximation for this 'n'.

Part 2: Spacing of Birthdays

Now, suppose we treat the year as a continuous interval [0, 1] instead of discrete days. We place n = 300 points uniformly and independently at random on this interval. Let X be the minimum distance between any two points. Calculate the expected value of X, denoted E[X].

Part 3: The Closest Pair

If we have 300 points on the circle of length 1, what is the probability that the distance between the two closest points is greater than some value 'd'? How does this relate to the Poisson process approximation for small 'd'?

Constraints and Assumptions

  • Assume birthdays are distributed uniformly across the year.
  • Ignore leap years (N = 365).
  • For Part 2 and 3, points are placed independently; the circular nature of the year should be considered.

Hints

Hint 1

For Part 1, express the probability of no collisions as a product and use the approximation 1 - x ≈ e^(-x).

Hint 2

For Part 2, consider the interval as partitioned into segments by the n points; think about the distribution of spacings between order statistics.

Hint 3

For Part 3, think of the points as events occurring in a Poisson process with intensity λ = n, and use the properties of exponential distributions to estimate the minimum gap.

Follow-up questions

  • How does the probability change if the birth distribution is non-uniform (e.g., more births in certain months)?
  • If we have a group of size 'n', what is the expected number of pairs that share a birthday?
  • How would you simulate this to verify the result for n = 300 in a programming environment?