You can't judge a slot machine by its belly-glass

Pretend you go to a casino and notice two slot machines side-by side. Hypothetical simplified configurations will be used as an aid in understanding. The devices seem identical, except that the returns shown on the belly glass for winning reel combinations with $1 bet are $1,000, $100, $10, and $1 on the first and $2,000, $200, $20, and $2 on the second. For reference, call the two machines Little Squirt and Big Shot, respectively. Do the obvious differences indicate which is the better game?

The answer is "no." The reason is that payoffs comprise only half the information you need to fully describe a gambling situation. What's missing is the prospect of achieving each result.

When games are machine forms of anything played with cards drawn from a standard source, pairs of dice, wheels like those at roulette tables, or some such, probabilities are predetermined. For instance, the chance of a six of spades as the initial card drawn from a single deck with no jokers can't be set at will. It's one out of 52 (1.92 percent). The likelihood of any other six as the second card after the six of spades has already been dealt is three out of 51 (5.88 percent). Similarly for dice, the chance of a three (1-2 or 2-1) is two out of 36 (5.56 percent).

On slots that don't mimic known phenomena, probabilities can be picked arbitrarily. Subject to some constraints, of course. These curbs normally comprise an edge that produces a target average rate of profit on the amount bet, and a hit rate and volatility that give players long and exciting runs for their money.

Each of the prototype machines might have the same 95 percent player payback, earning the joints the complementary 5 percent of the gross wager over an extended period. Possible, but not the only, sets of probabilities that yield this edge are as follows. On Little Squirt: $1,000 (0.05 percent), $100 (0.24 percent), $10 (1.22 percent), $1 (9.75 percent), and lose (88.74 percent). On Big Shot: $2,000 (0.02 percent), $200 (0.12 percent), $20 (0.61 percent), $2 (4.62 percent), and lose (94.62 percent).

These figures reveal two penalties for choosing the higher-payoff version. Corresponding combinations are roughly twice as hard to hit on Big Shot as Little Squirt. Also, the hit rate on Big Shot is about half that on Little Squirt, 5.38 versus 11.26 percent.

That's not all. The bosses could change the probabilities yet keep the same overall player payback percentage. As an example, say they assume solid citizens would be happier hitting more often. Then, for instance, they might set up Big Shot as: $2,000 (0.01 percent), $200 (0.19 percent), $20 (1.15 percent), $2 (6.89 percent), lose (91.76 percent). The jackpot will be less frequent than in the first-cited option, 0.01 as opposed to 0.02 percent (nobody would notice, odds being so slim either way). But lower-paying combinations will hit more regularly. The overall hit rate will be 8.24 rather than the previous 5.38 percent. Keeping bettors at the machine longer not only makes them happier, but should please the bosses, too. Taking the long-term perspective they were supposed to learn in Statistics 101, they know the average profit remains at $0.05 per dollar bet. Those who were ambitions enough to take Statistics 102 would also know they earn a greater profit by giving players more action on given stakes.

If the analysts cared to fool you, they could adjust the chances of Big Shot to increase the hit rate but lower the return percentage. The bosses get a double bonus and the players are pleased as well. Here's a set of probabilities that would cut the return to 90 percent, raising the casino's profit to an average of $0.10 on the dollar, while boosting the players' hit rate to 7.69 percent: $2,000 (0.01 percent), $200 (0.18 percent), $20 (1.07 percent), $2 (6.43 percent), and lose (92.31 percent).

None of this should necessarily stop you from trying a machine to which you're attracted. But all of it should imply that whenever you think you've got everything doped out, think again. For, as the punters' poet, Sumner A Ingmark, artfully articulated:

If in appearances you're a believer,
Know a casino can be a deceiver.