A group of $n \ge 4$ people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.).
(a) Let $I_{ij}$ be the indicator r.v. of $i$ and $j$ having the same birthday (for $i \lt j$). Is $I_{12}$ independent of $I_{34}$? Are the $I_{ij}$'s independent?
(b) Explain why the Poisson Paradigm is applicable here even for moderate $n$, and use it to get a good approximation to the probability of at least 1 match when $n=23$.
(c) About how many people are needed so that there is a 50% chance (or better) that two either have the same birthday or are only 1 day apart? (Note that this is much harder than the birthday problem to do exactly, but the Poisson Paradigm makes it possible to get fairly accurate approximations quickly.)
$I_{12}$ is independent of $I_{34}$, since knowing that persons 1 and 2 having the same birthday provides absolutely no information about persons 3 and 4.
However, $I_{ij}$ are not entirely independent, as knowing that $I_{12}$ and $I_{23}$ does tell us that person 1 and 3 must have the same birthday.
Checklist for the applying the Poisson Paradigm:
In [ ]: