Online casinos have taken the gambling world by storm, offering players the convenience of playing their favorite games from the comfort of their own homes. Among the most popular games in online casinos are slot machines and crash games. While these games may seem like pure luck, there’s a significant amount of mathematics working behind the scenes. In this article, we’ll dive deep into the mathematical principles that govern these games and explore how they affect players’ chances of winning.

Slot Machines: The Realm of RNGs and RTPs

Slot machines are the backbone of online casinos, attracting players with their exciting themes, bonus features, and potential for big wins. But have you ever wondered how these machines determine the outcome of each spin? The answer lies in Random Number Generators (RNGs).

RNGs are complex algorithms that generate a sequence of numbers that determine the symbols landing on each reel. These algorithms ensure that each spin is independent and random, providing fair gameplay for all players. The RNG constantly generates numbers, even when the machine is not being played, and the moment a player hits the spin button, the most recent set of numbers is used to determine the outcome.

Another crucial concept in slot machines is the Return to Player (RTP) percentage. RTP represents the theoretical percentage of wagered money that a slot machine will pay back to players over the long run. For example, if a slot machine has an RTP of 96%, it means that for every $100 wagered, the machine will pay back $96 on average. However, it’s essential to understand that this is a long-term average and doesn’t guarantee that you’ll win $96 every time you wager $100.

Mathematically, the RTP is calculated as follows:
RTP = (Total amount paid out by the machine) / (Total amount wagered by players) × 100%


  • Total amount paid out by the machine: $960,000
  • Total amount wagered by players: $1,000,000

RTP = ($960,000) / ($1,000,000) × 100% = 96%

Crash Games: Riding the Multiplier Wave

Crash games have gained immense popularity among online casino enthusiasts due to their simple yet thrilling gameplay. In these games, players place their bets, and a multiplier starts increasing from 1x. The goal is to cash out before the game “crashes,” as the multiplier can stop at any time, and if a player fails to cash out before the crash, they lose their bet.

The house edge in crash games is determined by the average multiplier at which the game crashes. This is calculated using the geometric distribution, which models the probability of success in a series of independent trials. In crash games, the probability of success (the game not crashing) decreases with each multiplier increment.

The expected value (EV) of a crash game can be calculated using the following formula:
EV = (Probability of cashing out at a given multiplier) × (Payout at that multiplier) – (Probability of the game crashing before cashing out) × (Bet amount)


Crash game house edge 2%
Average multiplier at which the game crashes 1.98x (1 / (1 – 0.02))
Player’s bet amount $10
Player cashes out at 2x

EV = (0.5) × ($20) – (0.5) × ($10) = $5

This means that, on average, the player would profit $5 for every $10 bet placed in this scenario.


While slot machines and crash games may appear to be based purely on luck, mathematics plays a crucial role in determining the outcomes and the profitability of online casinos. Understanding concepts like RNGs, RTPs, and expected values can help players make informed decisions when gambling online. However, it’s essential to remember that gambling should always be done responsibly and within one’s means. By grasping the mathematical principles behind these games, players can enjoy the excitement of online casinos while being aware of the risks and potential rewards involved.

By Michael