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?