Near-Birthday Problem: Birthdays Within One Day

Fourteen people have birthdays chosen independently and uniformly on a circular calendar of 365 days (day 1 is adjacent to day 365). Two people have a near-match if their birthdays differ by at most 1 day (i.e., they land on the same day or on consecutive days). Let $M$ be the number of unordered near-match pairs.

(a) Compute $E[M]$.

(b) Using a Poisson approximation for the probability that $M \ge 1$, estimate $P(\text{at least one near-match})$.

(c) Contrast with the standard birthday problem: for $n = 14$ people, what is $P(\text{at least one exact match})$?