gambling probability

Most Casinoz readers would agree that mathematics helps win at casinos, but not all regular gamblers bother themselves with a deep study of this complex science.

The average user can easily manage without mathematical formulas, complex calculations, and statistical data. It is hardly fair to accuse them of laziness or ignorance.

Gambling software developers provide information about their games' RTPs (the return to the player). Basic strategies for most blackjack, poker, and video poker variations can be found online. Betting systems and other recommendations are published in numerous thematic media, including Casinoz.

Armed with expert advice, even a person far removed from mathematics can win in a casino. In the case of online gambling, one can resort to cheat sheets if necessary.

However, advantage gamblers still recommend mastering at least the basic mathematical terms and concepts widely used in this industry.

The article discusses the odds of winning in popular casino games. We will try to convey the most important information to the readers in a maximally accessible manner by breaking it down into specific examples.

What Is Probability?

It is important to understand the basic principles. Let's start with the main term:

Probability is assessing the possibility of a particular event occurring.

In other words, it is an attempt to determine how likely a given event will happen.

In the probability theory, this indicator is expressed as a number between zero and one:

  • If an event never occurs, its probability is zero;
  • If it is guaranteed to happen, its probability is one.

Mathematicians adhere to this method, but ordinary users in everyday life may use other ways of expressing probability. They will be explained below.

old white and red dice

How to Calculate the Odds in Gambling

This task may seem incredibly difficult to an unprepared person, but the calculations will not be challenging in simple situations.

The standard formula is as follows:

Probability equals the number of favorable outcomes divided by the total number of possible outcomes.

Let's analyze it using a simple example.

You flip a coin while playing an ancient game called "heads or tails." There are two possible outcomes. There is one positive outcome. We divide one by two and get 0.5 (or 50%). That is the probability of winning in this popular gambling game.

Now let's consider a slightly more complex example from roulette.

You make a bet on the red color.

  • There are eighteen red numbers, which represent the favorable outcomes.
  • There are also eighteen black numbers.
  • You also lose if the ball lands on zero.

Therefore, the total number of possibilities is thirty-seven. 18/37=0.4864.

As you can see, the odds of winning in roulette are lower than in the classic coin game. The odds are no longer 50:50, but the payout for this bet is 1:1. This is how the casino gains a mathematical advantage over the customer, allowing it to stay profitable in the long run.

If you calculate the probability of winning for any bet in European roulette and then compare it with the payout multipliers, you will find the same house edge in all bets.

How Is the Probability of an Event Expressed?

We have already mentioned that in mathematical theory, the probability is represented as a number between zero and one. In everyday conversations outside the realm of science, other expressions are often used:

  1. Percentages - it's clear: 50% or 95%;
  2. Odds - negative and positive outcomes are juxtaposed: one-to-one, two-to-one, etc.;
  3. Fractions - the format is 1/3, 1/5, etc.

Here's an example. You're playing on a no-zero roulette wheel with thirty-six numbers. You place a bet on black. There are eighteen black numbers. The probability of winning is 18/36 = 0.5.

Alternatively, this indicator can be expressed as 50%, one-to-one, or ½.

Anticipating questions from players unfamiliar with no-zero roulette, we need to explain the following: payouts in this roulette are usually lower than in European one, or fees may be deducted from the winnings.

Another example. Suppose you're playing on a video slot and participating in a round where you must guess the suit of a card. There are four suits, but only one of them suits you. Accordingly: ¼ = 0.25.

In this case, the probability can be expressed as 25%, three to one, or ¼.

It's worth noting that these alternatives are not typically used by mathematicians in scientific literature.

Calculating Odds in Various Gambling Games

Now let's use some examples to understand how to determine the probability of multiple events simultaneously. We will calculate them considering various gambling games.

coin tossing probability

Head or Tails' Odds

Once again, the classic coin toss will come to our aid. The game is commonly known as "heads or tails." Let's calculate the chances of getting "heads" twice in a row.
Important rule:

To calculate the probability of two events, multiply their odds.

In this case, the events are independent of each other. In other words, the probability of each event is 0.5.

Therefore, we need to multiply ½ by ½, which gives us ¼ (0.25). These are the chances that the coin will land on the same side (heads or tails) twice in a row.

This can be easily verified by analyzing all possible outcomes of the two coin tosses:

  1. "Heads" can occur twice.
  2. "Tails" can occur twice.
  3. "Heads" can occur on the first toss and "tails" on the second.
  4. "Tails" can occur on the first toss and "heads" on the second.

Thus, there are four possibilities, which means the probability is 25% (0.25, 1/4, or one to three).

Odds in Six-Sided Dice

The same principle can be applied to calculate the odds in dice games.

For example, what is the probability of getting three twice in a row?

For a single event, this probability is 1/6. The rolls of the dice are independent of each other. Therefore, we multiply them together and get 1/36, or 2.7%.

Suppose we want to determine the probability of two events occurring simultaneously. For instance, we want to know the chances of rolling a three OR a six in a single dice roll.

In this case, we add the probabilities of both events: 1/6 + 1/6 = 2/6, or 1/3 (33.33%, 2 to 1, or 0.33).

playing cards and red dice

Odds in Card Games

In card games, the probability can be calculated using similar rules.

  • A traditional deck without jokers consists of fifty-two cards. The chances of drawing a specific card are 1/52.
  • There are thirteen card ranks: from two to ace. The probability of getting a card of any specific rank is 1/13.
  • There are four suits: spades, diamonds, clubs, and hearts. The odds of drawing a card of a specific suit are ¼.

However, it's important to consider the peculiarities of card games. During gameplay, cards are removed from the deck. They go to players, the dealer, or the discard pile. Each of these actions changes the probability of certain cards being drawn.

For example, there are four twos in a deck of 52 cards. The probability of drawing a two from a full deck is 4/52 or 1/13. Let's say a dealer deals a two on the first hand in blackjack.

What are the chances of getting a two on the second hand?

51 cards are remaining in the deck. Three of them are twos. Therefore, the probability of drawing a two is 3/51 or 1/17.

Here's another question:

What is the probability that three randomly drawn cards from the deck will all be diamonds?

The probability of drawing a diamond card on the first draw is 13/52. On the second draw, are 51 cards left in the deck, and twelve of them are diamonds (since one has already been drawn). Thus, the chances are 12/51. On the third draw, we have a deck of fifty cards, eleven of which are diamond cards. The odds are 11/50.

Next, we calculate:

(13/52) * (12/51) * (11/50) = 1716/132600.

Rounding it off:


In another expression, it's 0.01294 or 1.29%.

We determined the chances of finding three consecutive cards of the same suit using these calculations. The important condition is that we do not return the used cards to the deck at the end of every round.

Odds in Roulette

Let's move on to the "Queen of Casinos" – that's the grand title given to roulette by players.

Calculating the odds here is somewhat easier than in card games. We'll focus on European roulette with thirty-seven numbers ranging from 0 to 37.

The probability of winning when betting on a single number is 1 in 36 (1 out of 37). The payout for this bet is 1 to 35. This demonstrates the casino's mathematical advantage over the player, also known as the house edge.

The situation is similar to other bets in roulette. The payout multipliers for all bets ensure the casino's advantage over the player.

For more information on the house edge in different types of roulette, you can read thematic articles on Casinoz.

Odds in Blackjack

When discussing blackjack in the context of the topic at hand, it's crucial to note a significant characteristic of this game:

Every card on the table changes the player's odds of winning or losing.

Imagine playing with a single deck. To get a natural blackjack, you need an ace. There are four of them. If one ace comes out, the probability of obtaining the desired combination decreases by a quarter. And if all four aces have already been used, there is no chance of getting a blackjack.

During the game, cards are constantly drawn from the shoe. The pace of the gameplay can be very fast. Consequently, calculating the probability of various outcomes in blackjack is quite challenging.

Fortunately for novice casino customers, professionals have done the primary work for them. They have calculated how different rule aspects or game situations affect the RTP, house edge, and other parameters. Based on this data, tables of optimal strategies for the key game versions have been compiled.

Moreover, there are computer programs that allow for calculations in non-standard situations.

Probability in blackjack is a vast topic covered in numerous studies, articles, and books. Casinoz also addresses it in several pieces.

slot machine winning probability

Odds in Slot Machines

Calculating the probability in slots independently is impossible due to insufficient information. For example, you don't know how often symbols appear on the reels. You can study the paytable and find information about the RTP, but that's about it. Beyond that, you would need the assistance of a software developer.

To be fair, it should be noted that some providers make information about the symbol frequency, the occurrence of combinations, and other technical characteristics of slot machines publicly available.

This is informative, but it's challenging to say how a regular online casino visitor can practically use this data. It won't help them win. However, regular fans of video slots usually have little interest in such information.

Odds in Video Poker

Although video poker machines resemble video slots, they have much more in common with poker than with slot machines. During the game, the player must decide how to proceed in the current situation. Their actions directly influence the casino's advantage. In most slots, the outcome depends solely on the random number generator.

It's important to understand that video poker simulates a real deck of cards, and the random number generator plays the role of the dealer, shuffling and dealing the cards. Accordingly, you can calculate the probability of any card appearing or the chances of getting a specific poker hand.

Of course, there's no need to do this calculation alone. Mathematicians and professional players have done all the calculations, based on which they have developed optimal video poker strategies for all essential game variations.

Can Betting Systems Help You Beat the Casino?

Lastly, let's touch on a question directly related to probability in gambling.

Before providing an answer, we need to clarify the terminology:

  1. Betting systems are algorithms for increasing or decreasing the bet based on the previous round's results. The classic example is the Martingale system in roulette. They also include methods of changing bets on slot machines and similar techniques.
  2. Gambling strategies refer to making mathematically reasoned decisions. These decisions can reduce the house edge or give the player an advantage over the casino. For example, this includes basic blackjack or video poker strategies.

So, betting systems do not help you beat the casino, but strategies can be genuinely effective. In some gambling games, the client cannot influence the house edge; in others, the result directly depends on their actions.

This topic is covered in themed articles on Casinoz. You can find them in the Articles and Strategies sections.

Why Should You Know the Odds in Casino Games?

Probability theory is applicable in many aspects of gambling. Software developers for online casinos use it when creating new games. Professional gamblers rely on it when devising strategies.

However, it is important not to overestimate its significance for the average casino visitor. The capabilities of probability theory in a casino are not limitless. It won't help you beat roulette or win against a slot machine.

Be realistic. Casinos are designed for gambling operators to profit from customers. The mathematical advantage is almost always on their side. To stay ahead as often as possible, look for games with minimal house edge and learn to play them correctly. The articles on Casinoz can assist you with that.

You can put your acquired knowledge into practice at the well-known online casinos listed below. They offer interesting bonuses, numerous games across all genres, and excellent service.

Frequently asked Questions

👨‍🏫 Do you need to study probability theory to play in a casino?

No, the average player can do without this knowledge. Even professional gamblers don't need to be mathematicians to play successfully in a casino. All the necessary calculations have already been done. It's just a matter of using them correctly.

🎲 Are the results of individual rounds in gambling games connected?

No, in each coin toss, the probability of heads or tails is 50:50. It doesn't matter how many times one side or the other has appeared before.

👎 Do betting systems help beat the casino?

If you're referring to algorithms like the Martingale system, then no, they don't help you win in a casino. They don't reduce the casino's mathematical advantage over the player.

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