A word meaning random but restricted
Solution 1:
I suggest aleatory
in British English
- dependent on chance
- (esp of a musical composition) involving elements chosen at random by the performer
Collins English dictionary
Word origin C17: from Latin āleātōrius, from āleātor gambler, from ālea game of chance, dice, of uncertain origin suggest aleatory
Although the main meaning is similar to random the etymology of the word relates to chance selection from a limited number of possibilities (such as in dice, or cards) rather than selection from an unbounded set, and therefore suits your purpose.