The game of roulette has been on my mind lately, for no good reason, and I have been thinking of what it would be like to stretch the concept out mathematically by altering the number of pockets the ball could fall into for any given spin.transcendental? Of course, you couldn't build a physical wheel with enough pockets to accommodate all the transcendentals (or even the ones over a finite interval of the number line), but notionally you could rig something up which would choose a random real number by spitting out an infinite number of nonrepeating digits after the decimal point, and if it matches the one you had selected in advance, you get to win all of the bets placed. The house would have an edge corresponding to the zero and double-zero slots by taking all bets when a rational number comes up.
For it to be a sensible game of chance, you would want to be tossing your bet in against a pot of transfinite cardinality - either or more bettors placing finite bets, or else a denumerable number of bettors making transfinite bets would do well to make the pastime worth one's while. To keep things moving along, it would be well to have the random number generator spitting out digits at a faster and faster rate so that the process converges in a finite time, and I presume those betting would use similar random digit devices to select their own numbers to bet on as well. Rather than having something with an infinite number of lottery balls to construct their pick, or choosing one of the known transcendentals, or something, that is.
In the happy event that the house pays off on a bet, that person would have the privilege of figuring out what to do with a transfinite amount of winnings in such a way that does not destroy the economic basis of civilization.
At any rate, this scheme seems better than the opposite extreme: a roulette wheel with only a single pocket on it. Ugh, boring. Especially if you are the casino.