We have been racking our brains to try to create a formula that works for our game. Hopefully you guys can help.

The formula needs to make one card rarer than another when being collected online after clicking the Collect button.

Are you making a card game? Like Yu-Gi-Oh?

I remember a game of cards I was playing when I was young back in the 80's. It resembled Yu-Gi-Oh, only it was with vehicles. Each deck had a different theme. Cars, airplanes, Freight ships, trucks, motorcycles.
Each card was showing a vehicle and it's aspects.
For example, when it was my turn, I was asking for a specification among the 6 that were written on the card. I was asking for the Mph of the car and if the opponent had a more powerful card than mine, he could take the card.

So I would suggest something like that, rather than the rarest card.