Birthday problem where all days have different probabilities So, the site seems to be filled with birthday problem questions, but most presume uniform probability for all days as far as I can tell.
I am looking for a way to calculate the probability of no collisions in the set, decent approximations are good enough for this.
$p =(1-q)^{\frac {N^2}2}$ has previously been given to me as a way of approximately calculating this probability, where $q$ is the probability of a pair matching, and $N$ is the number of people, which seems simple enough where all days are equally likely.
My question therefore is, where the days all vary in their likelihoods, is $q$ simply an average of all the probabilities of a collision at each day, and does this seem a reasonable mathematical approach to take?   

If each birthday is selected at random using the same probability distribution, this is a reasonable approach.  The $q$, however, should be the chance of a match for any given pair, so should be the sum of the squares of the probabilities of each date.  So if each date has probability $r_i, i=1\dots n, q=\sum_{i=1}^nr_i^2$.  In the uniform case this gives $q=\frac 1n$, but in the nonuniform case it will be higher.  This explains Peter's first observation.
