Graphics
Visual component of modern single and online games are textures of high quality, animation, three-dimensional character models. 3D-modeling technology is developed taking into account mathematical concepts such as: vectors, matrices.

The knowledge is used by programmers to create realistic lighting, believable three-dimensional models of people, animals. Similar technologies are used in the development of video slots – entertainment at online casinos https://onlinecasinozonder.com/. Gambling games acquire features similar to the rating titles with multimillion-dollar budgets.

Artificial intelligence
The games use mathematical principles, as far as non-player characters – NPCs. This manifests itself in the gameplay when the player interacts with the NPC – controlled by a program. The laws of logic, as well as certain regularities, are the basis of the technology.

Features of non-player characters and computing algorithms:

-React to player action. Characters interact with the user: respond, perform an action. This is due to the laws of algorithms and logic.
-Scripts. NPCs’ actions are explained by the work of a script, which was originally programmed by the developer. This means that the character can not make an independent decision.
-Task. Characters serve a specific purpose in the game, as well as being primary or secondary.

There are non-player characters that are not controlled by the script, but by a representative of the development team. This approach is used in online projects.

Calculation and analysis of data
Mathematics in computer games remains an effective tool in the collection of analytical information. Data is needed to fix software bugs, bugs, improve the performance of the game on weak devices.

Mathematical models are added to automate the process of collecting information. This allows to speed up the work, as well as to avoid the drawback of the human factor. Modern technology is also used to analyze information when complaints and reports of different types are automatically categorized.

Developers thus work with the prepared material, which concerns the stability of the game on different systems. Calculations are based on basic mathematical laws. Without the use of formulas, the system would not work correctly.

Physics
Physical laws are used in most modern and classic game projects. This allows for a realistic gameplay experience. Most games with high scores on aggregators are made with an emphasis on realism – the laws of physics, nature.

Realism manifests itself in features such as:

-First-person shooters. Projects reliably recreated the principle of firearms and throwing weapons. This is due to the physical laws of motion, energy.
-Simulators. The most complex game projects, if we consider the issue in terms of physical laws. The authors are working to reliably recreate the movement of objects. Players are convinced of realism, which concerns the geometric shapes of objects: trees, mountains.
-The genre of sci-fi games. With the appropriate setting developers have to use the laws of physics, which are associated with the movement of objects in space. Accounted for the behavior of the body in weightlessness, the influence of gravity.

Physics in the games and refers to the destruction. There are several projects, which thoroughly implemented the destruction of objects: boxes, furniture, buildings, glass and other materials. The impact of a bullet of a firearm on an object is studied.

In racing physical laws are associated with damage to the car, with the impact of high speed on the object. In modern car simulators, vehicles “react” to the road surface, to the level of wheel inflation, to the weather conditions, and to the shape of the exterior elements of the car body.

Conclusion
Mathematics in games is the basis for the correct operation of scripts, item physics, and behavioral algorithms. Without the benefits of exact science, project development would take dozens of times longer, as well as require a lot of human resources. The use of mathematical formulas has made it possible to automate the process of improving games, eliminating bugs and errors.

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Before the mathematical concept of randomness was developed, people believed that everything that occurred was a result of the will of gods, supernatural entities that watched over human affairs and made decisions to “tilt the balance” in one direction or another to affect events. 

As a result, rituals and sacrifices were carried out to ascertain the “intent of the gods” or to try to have an impact on human affairs. This notion is still widely held, and many people still use lucky charms, follow superstitious customs, consult horoscopes, and hold some sort of conviction that there are such means of influencing their life. 

The Universe and the Earth

Other additional rituals attempted to combat nature’s and man’s condition’s randomness. Geomancy, the nine-square grid (also known as the magic square), and temple designs, the forerunners of board games, are a few of these that developed a particular mathematical interest.

Geomancy

Geomancy, which translates to “earth divination” or “divination of or from the earth,” is a scheme of 16 mathematically connected configurations of stones, beans, or other readily available tiny objects intended to make choices, provide answers to problems, or predict the future. The stones are thrown on the ground, and the resulting pattern is deciphered. Many binary “opposites” are represented by symbols, including male or female, good and evil, and sadness and happiness. Even and odd numbers can be represented by combining these opposites.

Each of these figures, like with all methods of astrology, has a variety of meanings based on how it relates to the other figures displayed and several other factors, such as the daytime, the climate, and the type of person posing the question.

The Nine-Square Grid

The 9-Square Grid is thought to have originated from an old method of land division, presumably from medieval India. The nine-square pattern, which in China consisted of eight farmer’s fields circling a central well, was thought to be the best layout. In Tibetan sacred diagrams, the grid of nine squares or a circle divided into 9 sections by parallel lines frequently occurs as the main form. 

In this way, the nine-square grid evolved from utilitarian uses in various cultures to the mystic significance and a portrayal of divine order and control by the gods.

Squares of Magic

They are directly connected to the Divine Grid, with the alleged numerical riddle that underpins their physical appearance being the Sacred Grid. The square of nine, which is attributed to Saturn, is the simplest magic square. Its rows and columns add up to 15, making it the square of nine; its diagonals add up to 30, and its rows and columns add up to 45. Jupiter is represented by the 4×4 square with columns and rows number 34, and Mars by the 5×5 square with columns or rows number 65.

Game boards

They are unmistakably connected to astrology, divination, and sacred geometries, and the patterns of the decks can reveal their esoteric or spiritual beginnings. The outcome of the dice-controlled game “snakes and ladders” now refers to good and bad “luck” as the game progresses; the stairs and snakes originally signifying good and evil fortune. Some of the designs on the boards resemble the layouts of cathedrals and holy places with a central “sacred area.”

Magic and Mathematics

Few people in the past could comprehend even the most basic geometry and arithmetic, and there is a long history of conflating mathematics with magic.

The positions of the planets and the precise times that astronomical phenomena appeared in particular parts of the sky might be predicted by those who were familiar with the regular motions of the cosmos. These highly competent workers, known as “priests,” worked in both the scientific and religious fields in earlier civilizations. After Christianity arrived in Europe, the religious component of these customs was derided as superstition.

These methods involved numbers, thus anyone who did so was viewed with a great deal of mistrust. Since the terms “astrologer,” “mathematician,” and “conjurer” were practically interchangeable, legitimate mathematicians were viewed with mistrust by the uninformed.

The Evolution of Probability

It’s thought that since cubes were used for gambling, religious rituals, and divination, persons who used them had a strong intuitive sense of the frequency of certain number combinations. The Latin poetry De Vetula, which depicts all the permutations for the collapse of three dice and is said to have been composed in the early thirteenth century, is the first printed example of the alternatives with three dice. Although it is mentioned in the poem, the use of binomial coefficients to determine the possibilities is not pursued until much later.

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A credible event is an event that, as a result of a test (the implementation of certain actions, a certain set of conditions) is bound to happen. For example, in terms of gravity, a coin tossed will certainly fall down.

An impossible event is an event that knowingly will not happen as a result of the test. An example of an impossible event: Under the conditions of terrestrial gravitation, a flipped coin will fly upwards.

Finally, an event is called random if, as a result of the test, it may or may not happen, and there must be a fundamental criterion of randomness: a random event is a consequence of random factors, the impact of which is impossible or extremely difficult to predict. Example: as a result of a coin toss an “eagle” will fall out. In the case considered, random factors are the shape and physical characteristics of the coin, the force/direction of the throw, air resistance, etc.

The underlined criterion of randomness is very important – so, for example, a card cheat can very cleverly imitate randomness and let the victim win, but we are not talking about any random factors affecting the final result.

Any result of a trial is called an outcome, which, in fact, is the occurrence of a certain event. In particular, when a coin is tossed, 2 outcomes (random events) are possible: heads will fall and tails will fall. Naturally, it is assumed that this test is conducted under such conditions that the coin cannot get on the edge or, say, hang in weightlessness.

Events (any) are denoted by capital Latin letters or by the same letters with subscripts, e.g.: . The exception is the letter , which is reserved for other needs.

Let us write down the following random events:

• A coin toss will result in an “eagle”;
• A roll of the die will result in 5 points;
• a card of the suit of clubs will be drawn from the deck (by default, the deck is full).

Yes, events are written in this way in practical problems, and it is convenient to use “talking” subscripts in appropriate cases (though one can do without them).

It should be emphasized for the third time that random events necessarily satisfy the above criterion of randomness. In this sense the 3rd example is again illustrative: if all cards of the clubs suit are initially removed from the deck, the event becomes impossible. On the contrary, if the tester knows that, for example, the queen of clubs lies at the bottom, he can make the event credible if he wants to =) Thus, this example assumes that the cards are well mixed and their shirts are indistinguishable, i.e., the deck is not mottled. And, here, “speck” doesn’t even mean “skillful hands” that eliminate the randomness of your winnings, but visible flaws in the cards. For example, the shirt of the queen of clubs can be dirty, torn, taped up with scotch tape… man, it’s some kind of manual for beginner chicatillos =)

Thus, when drawing an important lot it always makes sense to casually look to see if the faces of the coins are the same

Another important characteristic of events is that they are equally possible. Two or more events are called equiprobable if none of them is more possible than the others. For example:

A roll of heads or tails on a coin toss;
1, 2, 3, 4, 5, or 6 on a die roll;
drawing a card of clubs, spades, diamonds or hearts from the deck.

This assumes that the coin and die are homogeneous and geometrically correct, and that the deck is well mixed and “perfect” in terms of the indistinguishability of the card shirts.

Can the same events not be equally possible? They can! For example, if a coin or die has its center of gravity displaced, then quite certain faces are much more likely to fall out. As they say, another loophole for cheaters. Events-extracting clubs, spades, hearts, or diamonds are also equiprobable. However, equiprobability is easily broken by a trickster who, while shuffling a deck (even a “perfect” one), will cleverly peek and hide in his sleeve, such as the ace of clubs. Here it becomes less possible that the opponent will be dealt a club, and more importantly, less possible that he will be dealt an ace.

Nevertheless, in the three cases considered, the loss of equal possibility still preserves the randomness of events.

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WHAT IS PROBABILITY THEORY?
The result of research regarding the effects of randomness and uncertainty on social, behavioral, and physical phenomena is a branch of mathematics devoted to probability theory. Quantitatively, probability is defined by a number from 0 to 1, where 0 means the ultimate impossibility of an event and 1 is one hundred percent certainty that the event will occur. The more this number approaches 1, the greater the probability that certain events will occur. Probability is also measured on a scale of 0 to 100%.

A simple example of probability is a toss-up: a toss-up of heads or tails is equal in probability because there are no other outcomes for such a coin flip. In practice, probability theory is used to model situations where, under the same conditions, we have different results due to the same actions.

The outcome of a coin flip is random. Random events cannot be fully predicted, but they all have long-term patterns that we can describe and quantify with probability.

Let’s consider three basic theories.

Equally probable outcomes.
There is no reason to claim that the probability of one outcome of an event takes precedence over other outcomes. Imagine a vessel with identical balls that have been thoroughly mixed. The player is asked to take out one of the marbles, with the probability of choosing each of the marbles being the same. If a given situation has a number of results equal to n, then the probability of each result is 100%.

Frequency theory
According to this theory, probability is the limit on the relative frequency with which an event occurs under repeated conditions. The statement “the probability that A will happen is p%” in this case means the following: if you repeat the experiment over and over again, independently and under approximately the same conditions, the percentage of time that A will happen approaches p. The relative frequency is calculated solely after the experiments are conducted based on the data actually obtained.

If a number of experiments are performed under unchanged conditions, the relative frequency becomes stable, i.e., it varies within marginal differences. For example, a professional archer has fired 100 shots and hit the target 90 times out of them. His probability of hitting the target under certain conditions is 0.9. If he fired 10511 shots in his career, of which he hit the target 9846 times, the relative frequency is 9846/10511=0.9367. This figure will be taken into account to predict the archer’s result in future competitions.

Subjective theory
This type of probability is used in the decision-making process in order to further predict human behavior. It has no statistical characteristic. In this case, the probability is the level of verification of a certain statement. For example, the appropriateness of investing in various risky projects, participation in the lottery, planning drug stocks in medical institutions, etc. Subjective probability is determined by means of appropriate local expertise.

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Determine one event with one result. First, you need to determine the probability that you want to calculate. For example, you need to know the probability that a roll of the die will yield a deuce.

Find out the total number of scenarios that can occur. During step one, you determined the event. If you refer to the dice roll example, the total number of scenarios is six, because there are six numbers on the die. So a roll of two can have six different scenarios.

Divide the number of events by the number of possible scenarios. Falling two during the first die roll is one event. It turns out that the probability of falling two is 1/6, and the probability of not falling two is 5/6. This results in a 1/5 or 20% chance of getting a deuce on the first roll.

But how do you calculate the probability with multiple random events? Your steps are as follows:

Determine each of the events you will be working with. Say you need to find the probability of a four on each of two different dice.

Calculate the probability for each event separately. It will be 1/6. This will also allow you to determine the probability of a four falling on two dice at the same time.

Multiply all the probabilities. In our dice example, 1/6×1/6 = 1/36 is the odds that a four will fall on two dice at the same time.

If you find it difficult to figure out probability theory on your own, you can always ask a tutor. A professional tutor will show you how this theory works to solve real life and professional assignments. You will not only be able to discover this useful section of mathematics, but also to apply it to work and practical situations.

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