What Are The Odds?
Law: Most players don't know, but chances of hitting a jackpot go back to a 300-year- old mathematical principle.
Medicine & Science
Imagine this scenario: Jack plunks down a cup of quarters in front of a Pimlico slot machine, loses 50 times in a row and quits in disgust. As he stalks away, Jane spies his machine, drops a quarter in the slot, then laughs as she wins 100 coins.
Jack, of course, is convinced he cheated himself by quitting too early - and furious at the woman for taking "his" winnings. Now Jane drops 99 of her new-found quarters into the same machine, losing every time. But as she prepares to play her 100th quarter, she is convinced her odds are better than when she started. "That's a common fallacy, and it's completely wrong," says Michael Shackleford, a former Social Security actuary who runs a casino advice Web site called the Wizard of Odds.
According to Shackleford and other math Wizards, millions of players are ignorant of the 300-year-old mathematical principle that governs slot machines.
But that very ignorance will guarantee millions for Maryland and the racing industry - if Gov. Robert L. Ehrlich Jr. succeeds with a plan to put one-armed bandits at the tracks.
Discovered by Swiss mathematician Jacob Bernoulli in 1689, the principle is known as the Law of Large Numbers, and it has two parts.
First, it says, random outcomes such as a coin flip or tug on a slot machine lever are unaffected by previous flips and tugs. Each has the same odds as the one before it and the one after.
Second, the overall result of random outcomes generally becomes more predictable as the number of outcomes increase. This is known as the Law of Averages: It means that flipping a coin 10 times may yield nine heads and a tail, but flipping it a million times will yield close to 500,000 of each. "The key is that things that are unpredictable on a small scale are predictable on a large scale," said James Fill, a professor of mathematics at the Johns Hopkins University.
No one can say with precision just when the Law of Large Numbers kicks in. But Shackleford gave it a try two years ago when he dropped 4,000 quarters into the same Reno slot machine over 10 hours and recorded the results. At times winning and losing in the course of the day, he wound up short about $50 at the end.
"It really didn't take that long," he said. "I wanted to document the actual results."
If the General Assembly approves slots, the state will ultimately determine the results for Maryland's bettors by deciding what percentage of the cash gambled must be returned in jackpots. Typically, that would be about 80 percent.
The rest of the money, known as the hold or the take, would be split among the track owners, the horse racing industry and state and local governments. Once they know the ultimate return to bettors, casino operators can calculate their odds and jackpots - knowing that the outcome of millions of slot machine plays can be predicted with reasonable certainty.
So what about the odds on an individual machine?
That depends on the design. Slots have been around since the 1890s, and many still show bettors a display of three or more reels covered with symbols, such as cherries, lemons, bars and numbers.
The odds of winning vary with the number of reels, the number of symbols on each reel and the variety of combinations that win jackpots.
The math works this way: For a traditional three-reel, $1 machine with 20 symbols on each reel, if the casino wants only one winning combination - say three cherries - the probability of winning will be 1 in 8,000 plays. Theoretically, the house can make a $1 profit by offering a $7,999 jackpot on three cherries. That's not likely to be enough to keep a casino in business, though. If the house wants to keep 5 percent of the money wagered, it will make the jackpot $7,600 and keep $400, or 5 percent of the $8,000 wagered.
The odds could get longer if the machine has more reels or more symbols. For example, a machine with four reels and 250 symbols on each reel - and only one winning combination - would produce odds of 3.8 million to 1. Nobody would stick around for such a rare combination to come up, so casinos sweeten the deal by providing smaller jackpots for different combinations of symbols. It is the same as the Maryland Lottery paying lesser sums for hitting four or five numbers in the Lotto's six-digit winner.
So if the three-reel machine with 20 symbols had a lesser payout of $10 for one cherry on any reel, the odds of winning would improve to about 1 in 7. The computer-driven slot machines that have been popular since the 1980s offer a variety of games and winning combinations. Calculating their jackpots requires sophisticated mathematical analysis and, in many cases, computer simulations.
But experts say the same general rule applies to all slots. "The player has to understand that the casino has the edge," says Robert C. Hannum, a math professor at the University of Denver and consultant for slot machine manufacturers.
He said people often play slots knowing the odds are against them, but they chalk up their losses as the cost of entertainment. "It's like watching a sporting event, only there's a chance for them to win something," Hannum said.
Casinos order machines with the size and types of payouts they want. But that can vary with the state, casino and machine. Nevada, for example, allows casinos to hold up to 25 percent of the amount gambled, while New Jersey allows 17 percent. Proposed Maryland legislation specifies holds of 17 and 13 percent.
The actual odds and the house advantage are not posted on slot machines or provided to players. But according to results reported to state regulators, most casinos pay out more than the minimum percentage - the result of competitive pressure.
Shackleford said his research shows that Nevada slots pay out $85 to $98 of every $100 bet. But there's still a healthy profit because a busy machine can generate 800 spins per hour.
"They make money because of the huge volume of bets," he said. Because casinos are permitted to average the hold among their machines, they will set a few machines with "looser" or more frequent payouts to encourage playing, while others are set "tighter." While gamblers often talk about "lucky" machines, Shackleford said, the loose machines are paying out only slightly more.
Since the 1980s, manufacturers have produced video-based slot machines with sound and animation. Some allow multiple simultaneous games, while others are hooked up to networks, so a number of players playing the same game can build up "progressive jackpots" that top $1 million.
Unlike the system of wheels, cogs and gears that governed their forebears, today's machines rely on a computerized random number generator that determines where each reel stops. It selects stops for every reel the instant the lever is pulled - the image of spinning reels coming slowly to a stop is strictly for effect.
"These generators are spinning out numbers at the rate of thousands per second, so it's a case of pulling the lever at exactly the right instant, no matter who is playing," Hannum says.
So if Jane wins on her first pull, right after a man has lost 50 times at the same machine, her jackpot is strictly a random occurrence.
But try telling that to Jack.
Playing the game
- House advantage: The percentage of money wagered by gamblers that's retained by the casino.
- Loose machine: A slot machine that returns a higher percentage of money to players than the house average, it does not mean a machine is ready to return a large jackpot.
- Tight machine: A machine that returns a lower percentage to players than the house average.
- Progressive jackpot: A system that places a fraction of each wager into extra-large jackpot pool that increases until someone wins. Can operate across multiple machines or even several casinos.
Typical house advantage
- Keno (average) 27%
- Slots 5-10%
- Roulette (double zero) 5.3%
- Caribbean stud 5.2%
- Blackjack (avg. player) 2.0%
- Craps (pass/come) 1.4%
- Video poker 0.5-3%