The very first thing is that how do you model the number of objects of a rarity in the pack.

If you assume the 6 objects are independent to each other, a possible model is the multinomial model. Let be the number of objects of the 4 rarities respectively. Then

One good thing is that each of them has a marginal distribution of Binomial distribution, and thus the expected value will be just equal to, e.g.

I am not sure about the exact details, so perhaps you can elaborate more first, say by giving an example.