I have considered looking at showing only a small handful on the screen at once but you still need a way to pick from any of the 100's of cards in the deck, so being able to cycle through sets of 20-30 cards at a time say.

Not having this option means you reduce the chance of finding the required greens because they are in the deck not visible and not pickable from.

I've been thinking about this and I'm not sure that's necessarily true. It's pretty interesting statistically. E.g., if there are 100 cards total with 1 green, and you randomly present a batch of 10 on-screen, I don't think your odds change for selecting the green card. I think that shuffling the deck between draws and again presenting 10 cards to choose from, now from the 99 remaining wouldn't actually help or hurt your odds on the next draw. I think that if you always shuffle and then backfill the 10 while you can (while deck size permits), the odds don't change for the person drawing.

I could be wrong. I think I'm going to model this and see what happens. So more soon...