Slot Machine Hit Rate: Another Path to Gambling Self-Delusion

Most slot aficionados realize that every machine has a certain, albeit undisclosed, return percentage. This is the fraction of the money passing through the unit which is repaid to players over an extended period. It's the complement of the edge or house advantage on the game. That is, a 90 percent return means the house keeps 100 - 90 or 10 percent and has a 10 percent edge.

Few solid citizens, though, understand that every machine also has a specific and likewise secret hit rate. This is the fraction of rounds in which players receive returns. The machines call all returns "wins," but they may just be partial or whole refunds of the current bets instead of actual profits. And, even players aware of such an arcanum as hit rate generally don't know that any relationship between it and return percentage is arbitrary.

In the long run, measured in millions of rounds on typical machines, return percentage is the more important of these two quantities. In the statistically short span of a session, day trip, or fabulous casino vacation, hit rate matters more to players. But the difference is psychological rather than fiscal. Theories of hot and cold cycles, luck running in streaks, and other superstitions notwithstanding, hit rate takes the limelight because frequent goose eggs are discouraging while regular positive reinforcements, however small, are heartening.

Casino bosses who got A in high school algebra can play head games with bettors by tweaking hit rate independently of return percentage. For simplicity, consider a one-coin machine with a four-tier return -- 1, 2, 10, and 100 units. Multi-coin slots with lots of payout levels and big jackpots are analogous, except that "the long run," where departures from expected averages are insignificant, isn't hours or days but months or years.

Say that the supposed slot is stipulated to return 95 percent and hit 40 percent of the time. The accompanying breakdown gives two of the infinity of ways this unit could be set.

Chances of various returns for two possible machine configurations with 95% returns and 40% hit rates
Return	Machine #1	Machine #2
1	15.445%	31.778%
2	22.000%	5.500%
10	2.444%	2.444%
100	0.111%	0.278%

Players are more apt to be optimistic on #1 because so many payouts -- nearly half of all hits -- are for two coins while #2 emphasizes one-coin returns. This creates the impression that #1 is "looser." Yet the return percentages and hit rates are identical, and the jackpot -- the 100-coin payout in this instance -- is over twice as likely to occur on #2 as on #1.

Alternately, picture two outwardly similar machines, with the above return levels, but set as follows: #3 has 95 percent return and 35 percent hit rate; #4 gives only 90 percent return but hits at a 45 percent rate. Since the casino earns 5 percent less on #3, by normal monetary standards it's better for players. Still, a clever implementation of the higher hit rate, such as indicated in the accompanying breakdown, can fool folks into favoring #4.

Chances of various returns for possible configurations of machines with different returns and hit rates
Return	Machine #3	Machine #4
1	12.576%	20.122%
2	20.000%	22.389%
10	2.222%	2.487%
100	0.202%	0.002%

Not knowing the vital statistics, but acting on experience, those who try both machines would tend to find #4 more appealing. It offers slightly greater chance of two- and 10-unit payouts and lots more one-unit returns. Jackpots are not normally included in short-term experience criteria because they're rarely achieved. And, when they are, most people credit luck or gambling prowess, not the prize being more probable. The likelihood of a jackpot is low in either case, but 100 times less in #4 than #3. This effectively hides the edge where it's least liable to be noticed.

So, is your pet slot superior to the dog next to it? You'll never go enough rounds to tell. As the immortal Sumner A Ingmark said:

With pure conjecture on the line,
Your guess is just as good as mine.