Birthdays are not spread uniformly across the 365 days. Does the probability that two people in a room of 23 share a birthday rise or fall relative to the uniform case? Give the argument, not just the direction.

Birthdays are not spread uniformly across the 365 days. Does the probability that two people in a room of 23 share a birthday rise or fall relative to the uniform case? Give the argument, not just the direction.

Approach: Write the no-match probability for a group of size n as a symmetric function of the day probabilities, then show that function is maximised at the uniform vector using either a two-coordinate exchange or Schur concavity.

The match probability rises, since the uniform distribution minimises it and any real skew increases it. For the birthday problem with day probabilities p_1 through p_365, P(no match among n people) = n! e_n(p_1, ..., p_365), where e_n is the elementary symmetric polynomial of degree n. That polynomial is Schur-concave on the simplex, so it is maximised at the uniform point, and the no-match probability is therefore largest under uniformity. The two-coordinate version of the same argument is direct: shift mass between days i and j holding the rest fixed, and e_n changes by a term proportional to (p_i - p_j), so reducing any imbalance raises the no-match probability. Real birth data is skewed enough to move the 23-person match probability of 0.5073 by only a fraction of a percentage point.

Follow-up: How large must the skew be to bring the 50% threshold down from 23 people to 22?

Key concepts: schur concavity, elementary symmetric polynomial, birthday problem, uniform distribution.