The Mathematics Behind Online Slot Machine Games: Probability and Payouts

Anyone who’s been in a casino or played online casino games for much of their lives may tell you they’re all luck and no skill. And they aren’t wrong; any game whose mechanics revolve around a random factor boils down to chance. If you win, good for you; if you lose, tough luck.

That hasn’t stopped people from discovering the science—or, more promptly, the math—behind games like Katalogslot365 and others. For them, understanding how they work is key to winning big. Of course, luck still plays a bigger role, but math helps make the games less one-sided.

This post will delve deep into the math behind a staple of brick-and-mortar and online casinos: slot machines. The conditions for winning can’t get any simpler: match three or so symbols or match the jackpot-winning combination. But that’s where the simplicity ends.

Random number generation

You can’t have a serious conversation about slot machine math without talking about a piece of tech called the random number generator (RNG). Jokingly referred to as “RNGesus” by gamers, it’s a code that lets a system choose a value from a set of values. No specific conditions to meet; just picking out numbers at random.

But as far as online slots are concerned, RNG is a misnomer. For starters, the process isn’t really random because a human has programmed it to happen—and it’s impossible to get a computer to do something without programming. As such, it’s often referred to as a pseudo-RNG (PRNG).

Whether RNG or PRNG, the key takeaway from this is ‘a set of values.’ It may be hard to win against random results, but knowing that it doesn’t derive values from a bottomless pit is good enough to crunch some numbers.

Slot machines at a glance

Your chances in one slots game aren’t necessarily the same in another. The market is rife with slots of varying mechanics. Horizontal matches may no longer be the only way to win; today’s slots spin more symbols than before. Nevertheless, determining the probability of a win comes down to three key factors:

  • The number of possible combinations
  • The number of winning combinations
  • The odds of a winning combination occurring

To keep things simple, this explanation will use a standard three-reel slot machine containing 10 unique symbols per reel as an example. That’s a ten-symbol reel multiplied by itself three times, resulting in 1,000 possible combinations, including winning ones.  

Given that this is standard slots, it’s highly likely that the game only pays out on one horizontal payline. Ten unique symbols mean the machine can yield up to 10 winning combinations, with the odds of one occurring at 10/1,000 or 0.01%. It’s also likely that only one leads to the jackpot (e.g., three lucky sevens), so the odds for that are 1/1,000 or 0.001%. 

To give you a perspective, a mother is slightly more likely to give birth to identical twins than win the slots jackpot. Then again, the odds of winning the Powerball jackpot are far, far lower.

As mentioned earlier, today’s slot machines have more paylines and repeating symbols. Despite having more ways to win, the odds remain on the low end, starting from 1/5,000


The industry tends to boast about its games’ chances of a win by publishing its return-to-player (RTP) rate. According to the U.K. Gambling Commission, the RTP is the average payout based on tens or hundreds of thousands of games played. The larger the bet, the closer to the RTP the player can expect to get back.

Unfortunately, too many players mistake RTP for expected earnings per dollar wagered. Just because a slots game has an RTP rate of 90% doesn’t mean the player will recover USD$0.90 after betting a dollar every game. Chance is still a factor, and there’s the risk of a winless run.

But say you do win back USD$90 throughout your USD$100 run (assuming you bet a dollar per game). Diminishing returns can kick in once you run these winnings back into the machine, with returns and spins getting fewer each run until your wallet’s bone dry.

However, a slots game’s RTP is a crucial factor in developing a betting strategy. It begs the question, “What’s the best way to minimize my losses in a game with diminishing returns?”

Martingale vs. Kerry

An article recently published in Scientific American discussed two approaches to wagering on casino games: the Martingale system and the Kelly criterion. If these terms sound scientific to you, they’re principles hinged on game theory.

The Martingale system involves doubling your bet each time you lose and resetting it if you win. Below is a scenario of how it works.

  • Bet 1: USD$1
  • Bet 2: USD$2 (win USD$2, lose USD$1)
  • Bet 3: USD$4 (win USD$4, lose USD$3)
  • Bet 4: USD$8 (win USD$8, lose USD$7)
  • Bet 5: USD$16 (win USD$16, lose USD$15)

The system guarantees your initial winnings back, which is both an advantage and disadvantage. But in the case of slots, you can only grow your bets so much (experts recommend spending no more than 16 times the first bet) before it becomes too risky. If the run doesn’t result in a single win, it’s better to swallow the USD$31 loss and start over.

Some experts believe that the Kelly criterion is more effective in mitigating a player’s losses, as it takes the odds of winning and losing into account. It involves an equation where the outcome is the ideal bet amount, which is a fraction of the player’s bankroll.


Make no mistake: this is a lot of math for a game with a straightforward way to win. It wouldn’t be surprising if you don’t like math. Regardless, the information can be valuable in making key decisions as you play slots or other games of luck. 

But above all else, it’s still important to know when to stop. 


Written by Austin Crane

Austin is the principle web director for Untamed Science and Stone Age Man. He is also the web-director of the series for the High School biology, Middle Grades Science and Elementary Science content. When Austin isn't making amazing content for the web, he's out on his mountain bike or in a canoe.

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