Dice/coin toss/random number simulator | ZeroTools

The core of this tool is a highly accurate numerical generation function that uses the Web Crypto API, a cryptographic pseudo-random number generator.

Rather than relying on standard JavaScript functions, it obtains a random number sequence that is difficult to predict and has extremely low bias by calling cryptographically secure functions.

Based on this API, we are building a mathematical model that determines the outcome of polyhedral dice, from tetrahedral dice to centahedral dice.

Multiplying a die with N sides by the obtained random floating point number U, taking the floor function of the result, extracting an integer value from zero to N minus one, and finally adding one, we can derive an exact integer from 1 to N with uniform probability.

This mathematical structure is fully extensible not only for hexahedral dice, but also for any special shape dice, such as icosahedral dice or centahedral dice.

In addition, when determining whether a coin is flipping (heads or tails), the software reproduces the simplest form of the binomial distribution, which determines one of two states with an exact probability of 50/50, using operations equivalent to dihedral dice.

Because the quality of entropy in random number generation comes from the hardware random number seed, it returns sufficiently reliable results even in environments that require statistical rigor.

Multiple random number distribution shapes can be selected depending on the user's purpose. Uniform distribution of specified minimum and maximum values ​​in a closed interval is achieved by applying the basic algorithm described above and multiplying the interval width by a random number.

On the other hand, a mathematical method called Boxmuller transformation is used to generate random numbers that follow a normal distribution, which is essential when simulating the natural world and statistical phenomena.

A random variable that follows a standard normal distribution is calculated using logarithmic and trigonometric functions from two independent uniform random numbers, and then scaled to the desired mean and standard deviation to generate a bell curve-shaped random variable that has a peak in the center and spreads to both tails.

Furthermore, support for discrete probability distributions such as the unomial distribution is incorporated as an internal calculation, making it possible to simulate complex probability models in real time, such as calculating the number of successes when repeating multiple independent trials with a specified success probability.

As a result, users can freely obtain not only random numbers with equal probability, but also random number sequences of arbitrary range-specified integers that meet more advanced mathematical requirements, such as biased events and events with cumulative probability fluctuations.

As visual feedback for random number generation, this simulator provides dice rotation animation that makes full use of the three-dimensional transformation function of cascading style sheets.

A three-dimensional polygon is constructed on the browser by combining transition and transform properties and placing the faces of each die at appropriate coordinates and angles in three-dimensional space.

When running a rotation animation, the rotation angles for each axis are dynamically calculated and applied so that the pre-calculated target face faces forward.

At this time, by setting easing using the cubic Bezier function, we reproduce the physical behavior of rolling, where the initial speed is high and gradually decelerates to a stop.

Furthermore, in order to express the uncertainty of rotation due to random numbers, we programmatically add a random number of spins and temporary axis wobbling in the process of reaching the final stopping angle, visually conveying the unpredictability of random numbers to the user.

This drawing process does not use heavy graphics APIs and is performed only with hardware acceleration optimized for the browser's native drawing engine, resulting in a fast and smooth visual experience that is independent of device performance.

The generated random number data and operation history are asynchronously persisted through the browser's standard local storage API and indexed database.

Results from thousands to tens of thousands of trials executed in a single session are immediately structured as serialized data in Jason format and stored securely in client-side storage.

This mechanism allows the previous trial history and random number distribution data to be completely restored and the simulation to continue even in the event of an unexpected interruption such as a page reload or browser restart.

In addition, the data stored includes not only the generated numbers but also all the metadata necessary for statistical analysis, such as the type of random numbers used, distribution settings, timestamps, and hashes of seed values.

In order to optimize memory consumption, we have introduced a circular buffer algorithm in which the oldest historical data is garbage collected first, and we have established a robust memory architecture that prevents performance degradation and browser crashes due to infinite data accumulation.

Accumulated trial data is linked with a data visualization library and projected on the screen as intuitive graphics. The probability distribution histogram is a powerful analysis function that aggregates the frequency with which each numerical value is generated by class and visually compares the difference between the theoretical value and the measured value.

The height of the histogram bar is dynamically updated in proportion to the number of trials, and as the number of trials increases according to the law of large numbers, you can observe in real time the process of convergence to the configured probability distribution shape.

In addition, the cumulative number of trials graph represents the number of times a specific condition was met and the cumulative sum of results as a line graph along the time axis.

This makes it possible to analyze time-series trends such as bias in random numbers and the frequency of successive occurrences of the same event.

All of these graphs are drawn using scalable vector graphics, so the resolution does not deteriorate even when scaling, producing high-quality images that can be used directly in reports, papers, etc.

Users can explore data in detail through interactive operations such as filtering displayed items and zooming in on the time axis.

The range of applications for this simulator is wide-ranging, from dice judgment in tabletop role-playing games to academic experiments that handle complex random variables, and even debugging work in game development sites.

In actual table talk role-playing games, even in environments where physical dice are not available, judgments based on completely random numbers are instantly executed and the session progresses smoothly.

In particular, percentage rolls using centahedral dice and complex damage calculations involving rolling multiple dice at the same time can be solved in an instant using the mathematical model of this tool.

In addition, for stochastic experiments, it provides a robust random number sequence that can be used in statistics and mathematics education, such as approximate calculation of pi using the Monte Carlo method and verification of the probability of event occurrence under specific conditions.

As a practical guide to debugging in game development, it serves as a powerful testbed for verifying whether the lottery logic of item drop rates and enemy behavior patterns follows the intended probability distribution.

By using this tool to instantaneously perform tens of thousands of lottery trials and check the results in a histogram, developers can quickly discover bugs and biases in the random number generation algorithm and obtain objective indicators for quickly and accurately adjusting game balance.