What is The Monte Carlo Simulation?

The Monte Carlo simulation is a probabilistic model that can include an element of uncertainty or randomness in its prediction. When you use a probabilistic model to simulate an outcome, you will get different results each time. For example, the distance between your home and office is fixed. However, a probabilistic simulation might predict different travel times by considering factors such as congestion, bad weather, and vehicle breakdowns.  

In contrast, conventional forecasting methods are more deterministic. They provide a definite answer to the prediction and cannot factor in uncertainty. For instance, they might tell you the minimum and maximum travel time, but both answers are less accurate.  

Benefits of the Monte Carlo simulation

The Monte Carlo simulation provides multiple possible outcomes and the probability of each from a large pool of random data samples. It offers a clearer picture than a deterministic forecast. For instance, forecasting financial risks requires analyzing dozens or hundreds of risk factors. Financial analysts use the Monte Carlo simulation to produce the probability of every possible outcome. 

History of the Monte Carlo simulation 

John von Neumann and Stanislaw Ulam invented the Monte Carlo simulation, or the Monte Carlo method, in the 1940s. They named it after the famous gambling location in Monaco because the method shares the same random characteristic as a roulette game.

The basic principle of the Monte Carlo simulation lies in ergodicity, which describes the statistical behavior of a moving point in an enclosed system. The moving point will eventually pass through every possible location in an ergodic system. This becomes the basis of the Monte Carlo simulation, in which the computer runs enough simulations to produce the eventual outcome of different inputs.

For example, a six-sided die has a one-sixth chance of landing on a specific number. When you roll the die six times, you might not land the die on six different numbers. However, you will achieve the theoretical probability of one-sixth for each number when you continue indefinitely rolling. The result accuracy is proportional to the number of simulations. In other words, running 10,000 simulations produces more accurate results than 100 simulations. 

The Monte Carlo simulation works the same way. It uses a computer system to run enough simulations to produce different outcomes that mimic real-life results. The system uses random number generators to recreate the inherent uncertainty of the input parameters. Random number generators are computer programs that produce an unpredictable sequence of random numbers. 

The Monte Carlo simulation compared to machine learning 

Machine learning (ML) is a computer technology that uses a large sample of input and output (I/O) data to train software to understand the correlation between both. A Monte Carlo simulation, on the other hand, uses samples of input data and a known mathematical model to predict probable outcomes occurring in a system. You use ML models to test and confirm the results in Monte Carlo simulations.

Probability distributions are statistical functions that represent a range of values distributed between limits. Statistics experts use probability distributions to predict the possible occurrence of an uncertain variable, which might consist of discrete or continuous values. 

Discrete probability distribution is represented by whole numbers or a sequence of finite numbers. Each of the discrete values has a probability greater than zero. Statisticians plot discrete probability distribution on a table, but they plot continuous probability distribution as a curve between two given points on the x-axis of a graph. The following are common types of probability distributions that a Monte Carlo simulation can model.

Normal distribution 

Normal distribution, also known as the bell curve, is symmetrically shaped like a bell and represents most real-life events. The possibility of a random value at the median is high, and the probability significantly decreases toward both ends of the bell curve. For example, a repeated random sampling of the weight of students in a particular classroom gives you a normal distribution chart.

Uniform distribution

Uniform distribution refers to a statistical representation of random variables with equal chance. When plotted on a chart, the uniformly distributed variables appear as a horizontal flat line across the valid range. For example, the uniform distribution represents the likelihood of rolling and landing on each side of a die.

Triangular distribution

Triangular distribution uses minimum, maximum, and most-likely values to represent random variables. Its probability peaks at the most-likely value. For example, companies use triangular distribution to predict upcoming sales volumes by establishing the triangle's minimum, maximum, and peak value.

These are two common challenges when using Monte Carlo simulations: 

• The Monte Carlo simulation is highly dependent on the input values and distribution. If mistakes are made when electing the input and probability distribution, it can lead to inaccurate results. 

It might take excessive computational power to perform Monte Carlo experiments. Computing with the Monte Carlo method can take hours or days to complete on a single computer.