It's a small thing, but I don't think it would be arbitrary to ask that dice-rolling be made into a simple game.
I have one idea: every die you get can be rolled two or three times (before it perma-rolls to one). So, if you get something less than six, you can ponder whether it's worth another roll. (For instance, let's say my first study insert's a four. I can decide whether that's what I want, or whether I want to try for something else. If I roll, I have a chance for something better. If I don't get a good roll second time around, that's my own fault, and that's OK. It's not merely fate.)
Another variant might be: I can roll the die in my inventory as many times as I like - until it lands on a "one". (So an example roll might be: 2, 3, 3, 5, 2, 1. Another might be 2, 3, 4, 3, 5. Or you could, theoretically, roll a round like this: 2, 5, 4, 6. And so on.)
The chance to get a six at the risk of getting a one would make playing with dice much more interesting, I think.