Are There Limits to Profits at the Slots?

How much can gamblers earn at the slots?

Those who relish the action ogle the payout lists on the faceplates. They figure the answer is the value of the jackpot: millions on some machines, thousands on others.

Those who fear the house advantage cringe at warnings about edge. They figure the answer is zero: the casinos ultimately get all the money and solid citizens pretend they had a good time anyway.

Reality lies in between. Fortunes fluctuate, up and down, as the inherent volatility of a game is superimposed on the gradual bankroll erosion caused by the edge.

There's no way to predict particular answers for particular players over particular periods. But the pervasive laws of probability, and a little math, define theoretical limits on the profits players can expect to earn. These limits aren't targets for sessions of reasonable length, any more than the speed of light is the goal of a rocket engine designer. Rather, they're useful to show the importance to players of factors such as individual payout levels and overall machine return percentages.

Say a casino has four types of slot machines. They all have 10 "equitable payout" levels, such that chances of winning are in proportion to the returns, subject to the casino's edge.

• Models A95 and A99 provide 95 and 99 percent returns, respectively, over extended periods. On both models, payouts (returns per coin bet) follow the casino's "A" schedule:
  1, 2, 5, 10, 25, 50, 100, 250, 500, 1000.
• Models B95 and B99 provide 95 and 99 percent returns, respectively, in the long run. On both models, payouts follow the casino's "B" schedule:
  1, 2, 5, 10, 25, 50, 75, 100, 250, 2500.

Contrast the machines two ways. (1) The two "A" models have the same payout levels but differ in return percentages, "99" being more favorable to players than "95." Likewise for the "B" models. (2) The two "95" models have equal return percentages but differ in their payout levels, schedule "B" being more skewed toward the high end than "A," with a bigger jackpot and smaller intermediate payouts. Ditto for the "99" models.

Theoretical profit limits for each machine can be calculated from the payout lists and the chances of a hit at each level. The table below gives limiting profit per dollar bet, above which fewer than five out of a hundred players can expect to go. The data also show the number of rounds required to reach the statistical maximum in each case.

The tabulated figures emphasize the importance of return percentage. Going from 95 to 99 percent return raises theoretical profit limits significantly. Skewness also matters. The more skewed "B" slots have higher profit limits than the corresponding "A" models.

Numbers of rounds to reach the maximum is high in all instances. This suggests that fluctuations associated with individual hits are more significant than house advantage during specific sessions of reasonable length and only diminish in impact after greatly extended play. The short-term consequences of volatility can be seen to be strengthened when return percentages increase and also as payouts become more highly skewed.

As always, a caveat. Theoretical maximum losses rise with numbers of trials just as do profits, but at three times the rate. For the same degree of confidence -- the level below which fewer than five out of a hundred players will drop -- the loss limit is therefore triple the corresponding win. So, weigh wisely the wary warning importuned in "Icarus" by the immortal Sumner A Ingmark: