How casinos choose payoffs: Ranked hands with elective probabilities

Casinos set returns on bets -- primarily to achieve a desired edge, and secondarily to get payoffs in round numbers for the convenience of all concerned. On wagers with unalterable probabilities, such as those at craps, options are limited. For instance, the odds against winning a Place bet on a 10 are 2-to-1. Paying 2-to-1, $10 for a $5 bet, the house would have no edge. Paying $9 for $5, house advantage is 6.7 percent. At $8 for $5, it would be 13.3 percent -- so high that players would stay away. Changing the base to $6 and paying $11 would be technically feasible, with 5.6 percent edge. But this would confuse the game too much. So the casino is pretty much locked into $9 for $5.

In games like video poker or Caribbean Stud, outcomes are more complex than win-lose-push. Some are tougher to get than others. And the games are designed so the less likely a result is to occur, the more it pays. The casinos can't adjust the chances associated with the various hands. However, they have substantial latitude in setting what they pay. The bosses can specify any edge they want, then calculate alternate payoff schedules to fit.

There's one more common class of games. On slot machines, probabilities are arbitrary. That is, not preordained by such things as numbers of grooves on a wheel, ways two six-sided dice can total 10, or combinations of the 52 cards in a standard deck that form a full house. Here, casinos have even more elbow room. In addition to edge, they can specify the overall "hit rate" for the game. With these values, they can then calculate sets of probabilities and payoffs for various results that do the job.

Here's a simplified illustration. Say a slot machine has four winning combinations stars, bars, bells, and whistles. The bosses want an edge of 4 percent (96 percent player payback). And they figure that a 42 percent hit rate will provide strong reinforcement. The accompanying table shows three out of the innumerable statistically-correct combinations of probabilities and payoffs that produce the target performance.

	machine 1	machine 2	machine 3
	chance	return	chance	return	chance	return
stars	0.128%	100	0.010%	150	0.011%	200
bars	1.833%	9	3.099%	10	3.022%	10
bells	13.347%	3	12.308%	3	12.334%	3
whistles 26.693% 1	26.583%	1	26.633%	1

This example shows that the casinos can do more than just juggle payouts and probabilities in games of this type. It also suggests that they can tweak both the hit rate and the edge at will.

Still, the casinos can't merely pull numbers out of the air. If they wanted a 50 percent hit rate, they could return 1-for-1 on every spin, or give winners back less than they bet. At 1-for-1, the payback rate would be 100 percent. The casinos not only wouldn't be making any money, but the game would be about as exciting and potentially profitable for bettors as making calls on pay phones to numbers they knew would be busy. As for the option of losing money on winning hands, the smoke and mirrors of multi-line multi-coin slots notwithstanding, solid citizens wouldn't beat down any doors to participate. Losing by winning would be even more of a problem with hit rates over 50 percent.

The 42 percent hit rate in the hypothetical example is high, and you can see that more than half of the "wins" are returns of 1-for-1 where players really only push because they get their money back. Lowering the hit rate accommodates more wins at higher levels, but it can lead to disheartening cold streaks.

The art and science of slot machines lie in choosing edge and hit rate, and -- with these -- schedules of probabilities and payoffs, as compromises. Values that give the house reasonable earnings and patrons gambling experiences they find worth what they risk and may lose. Unless, of course, what draws you to the slots are endlessly repeated clips of things like Blondie Bumstead, Lawrence Welk, or Pac-Man. In which case the immortal Sumner A Ingmark may have had you in mind when he wrote: E'en the wisest act like fools, When in quest of gold and jewels.