I highly doubt anyone has done enough sampling in this game to give you an accurate answer. Without passes, I believe there is a 0.1% chance of a rainbow pet, or 1 in 1000.
So, to simplify, every time you open an egg, the game randomly generates a number between 1 and 1000. If the number is one, the pet is a rainbow pet. But the key thing to remember is random.
The first egg you open could be rainbow. Or the 10th, or the 100th. Or, sadly, maybe the 1500th. It’s random, and so on average it should take you about 1000 times, but some will get it sooner, some much sooner, some later, and some much later.