Monte Carlo Simulation – Explanation, Examples, and Uses

monte carlo simulation

The Mathematician, Stanislaw Ulam, used a method that applies to card games. The technique is of inferential statistics and sampling randomly. It is the Monte Carlo solitaire he used in the Manhattan Project. Decision making is an essential part of every situation. To make a decision, one needs to think about all the probabilities. Every choice has two sides, like the pros and cons. One can face difficulties, uncertainty, variability, and ambiguity, and it means analyzing the risk of decisions someone is about to make plays a particular and essential role.

The real problem occurs in predicting the future. No matter how much information you have and how accurate they are; you can never predict what is going to happen. With the Monte Carlo method, it is possible to see all each possible outcome, which helps people to think more before taking a step towards something.

Monte Carlo is a method that works based on computerized math. It accounts for all the possibilities, positive or negative, about a decision. This method helps people to make decisions by analyzing risk factors. The person applying this method gets to see all the possible outcomes of an action. The scientists involved in the making of the atom bomb used this method for the first time. It got the name Monte Carlo from the renowned town of casinos, “Monaco.”

The introduction of the Monte Carlo Method happened during the Second World War. Since then, people use this method in almost every field that is disparate like engineering, finance, energy manufacturing, transportation, project management, environment, etc. It is the model method to show conceptual systems and different types of physical systems. Monte Carlo Simulation is a beneficial and famous mathematical technique. To know more about it, continue reading the whole article


Working principles of Monte Carlo simulation

In the Monte Carlo Method, it does the substitution of values of a range to make models of possibilities that can occur after deciding on something. This method uses various sets of importance according to the number of uncertainties and calculates results again and again for each probability to analyze risk factors. Using the Monte Carlo Method can show thousands to ten thousand outcomes to complete the calculation and gives possible outcome values.

Variables have various outcomes depending on their probabilities while using a probability distribution. In a risk analysis of variables, probability distribution describes uncertainties in a way that seems to be more realistic than others. Here are some characteristic phases of the probability distribution

Types of Distribution including in Monte Carlo Method

In the Monte Carlo Method, there are iterations, which mean a set of samples. From input distribution, a sample contains random values. It records the outcomes. Monte Carlo simulation gives approximately accurate results because it does this procedure thousands of times to present all possibilities. It describes the opportunities for action as well as the way something can occur.

Normal Distribution

Another name of this kind of distribution is the bell curve. In this method, the user gets a variation description about mean by defining the mean and the standard deviation. People’s height is one of that natural phenomena; the bell curve can describe. It is symmetric

Example: The inflation rate is an example of variables that a normal distribution can explain.

Lognormal Distribution

The lognormal distribution represents the values that have unlimited positive potential but doesn’t get lower than zero. It shows positive skewed values

Example: amounts of real estate property, oil reserves, etc. These are the examples of variable, lognormal distribution describes.

Discrete Distribution

The likelihood of a specific number that the user defines is the discrete distribution.

Example: the lawsuit is an ideal example of discrete distribution. In a trial, the probable chance of a result is

Prospects of the contrary verdict- 30%

Chances of favorable Verdict- 20%

Hopes of the mistrial – 10%

Chances of settlement – 40%

Triangular Distribution

In triangular distribution, the user needs to define three different values – minimum, medium, or most likely and maximum and occurring chances of getting a benefit from around the most probable benefits.

Note: triangular distribution describes variables that have inventory levels and past sale history per unit time.

Pert Distribution

This distribution method works the same as triangular, which means the user has to define the minimum, medium, or most likely and maximum value. Most likely, costs have more chances to occur, but in this case, the benefits in between the maximum and most likely have more occurrence chances than that of in a triangular distribution.

Example: Users use pert in project management models. It describes the duration of a task in the project.

Uniform Distribution

It is evident by the name of this distribution that shows uniform chances of occurrence. The person using this distribution must define the minimum and maximum values.

Example: With uniform distribution, users can describe the values of the manufacturing cost of any product or the future sales revenue of a project.

There is another effective method to calculate probabilities, and that is the single point estimate. But in the risk analyzing factor, the Monte Carlo simulation has got more advantages over this unique point estimate method.

Advantages of the Monte Carlo Method

  • Monte Carlo is a method that shows the relationship between each variable. It shows when some samples go up. Others get down. That is how it gives proper outcomes by displaying the input correlation.
  • Graphing is essential to know higher and lower chances of a consequence. With the Monte Carlo simulation, the result it reveals, creating a graphical representation, becomes more comfortable. It makes it simple and easy for analysts to communicate with other stakeholders about findings.
  • When it comes to combining various values of various inputs and building a model out of it to see how the scenario changes, it becomes difficult for users. However, the Monte Carlo method makes it possible to present the inputs with the same values in inevitable outcome occurrence. It has no value to analyze further.
  • Monte Carlo method describes what could happen after taking action as well as how likely the outcomes are.
  • Making a model requires information about the variable that has more impact on the findings. Monte Carlo method shows the input with the most effective results.
  • There is a use of hypercube sampling in the Monte Carlo method, which enhances it’s the accuracy of sampling from the whole input distribution range.

The procedure of the Monte Carlo Method

The entire monte Carlo method has three following steps

  • There are some input variables, and the first step is to sample randomly on them.
  • Building model and evaluating the output of the model
  • Make a statistical analysis of the model output.

Monte Carlo Method In Daily work

[email protected] with Monte Carlo Simulation

Professionals have this opportunity to apply the Monte Carlo method in the spreadsheet for analyzing practices. The dominating tool to analyze spreadsheets is Microsoft Excel, and with palisade @risk, it enhances the accuracy of the Monte Carlo method in excel. From accuracy in calculations to easy to use chance has a significant market value. Analyzing uncertainties and risk factors of large projects is the logical application Monte Carlo method, the introduction of Microsoft project led it

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Trading Risk analysis With Monte Carlo method

Most of the time, people used to use the strategy of understanding the history of market data to analyze the risk and find out if the plan would work in that case or not. This method seems helpful, but there is no certainty of the future. Just because something worked in the past situation doesn’t mean they will work in the future as well.

Most importantly, conditions keep changing from time to time, so fate will never be the same as the past. To overcome this, drawback professionals started using the Monte Carlo method to analyze risk in trading. In the Monte Carlo method, it generates a trade sequence by sampling the list of trades or trades distribution. After sampling, it analyses every series, and it gives proper results that determine the probabilities of each outcome.

Frequently Asked Questions About The Monte Carlo Method

  1. What is the definition of the Monte Carlo Method?
  2. Monte Carlo is a method to analyze risk factors or uncertainties of a specific system by generating random variables and sampling them.
  3. What is a standard example of the Monte Carlo Simulation?
  4. Rolling dice is a common phrase in sums that require probability—an amount of throwing two dice that have values one through six, respectively. Calculating the likelihood of this sum is an example of the Monte Carlo method.
  5. How much accuracy expectation the Monte Carlo Method carries?
  6. Considering any random factor that has three elements, which are error, the accuracy of the Monte Carlo Method in such cases is about 4 percent.
  7. Explain how the lawsuit is an example of the Monte Carlo Method?
  8. Prospects of the contrary verdict- 30%
Chances of favorable Verdict- 20%
Hopes of the mistrial – 10%
Chances of settlement – 40%
Monte Carlo Method includes discrete distribution. It is an example of the placement where it samples random values from input.
  1. Which software has the best opportunities for the Monte Carlo method?
  2. There are two monte Carlo method software, palisade @risk, and the Oracle crystal ball. Both of these are the best.
  3. How many iterations do a Monte Carlo Method can make at one time?
  4. The iteration between 100000 to 500000 times, results in ethical values. There could be 100000 iterations for hours. It depends on how complicated the algorithm is and how complicated the software is using to run the program.
  5. What should be the first step to use the Monte Carlo Method to analyze?
  6. The first step is the most crucial. There is a model that contains the variables, and one needs to find out all the positive, negative, most likely estimates for those variables present in the model.
Q Who invented The Monte Carlo Method?
  1. Stanislaw Ulam invented the mathematical technique to sample random variables from an input probability distribution and analyze risk factors.
  2. Where did Stanislaw Ulam contribute to this Monte Carlo Method?
  3. Stanislaw Ulam contributed his Monte Carlo Method in The Manhattan Project. He was working on this project with John von Neumann during the Second World War.
  4. Purpose of The Monte Carlo Method
The Monte Carlo method works excellent to build models of probabilities of different results. It is not easy to predict because of the random variables that keep interventing. Monte Carlo Method makes it simple and more comfortable to represent each factor that is possible about an action. It does not only show what the occurrence is going be, but it also shows how likely it is. From finance to engineering and project management, all these fields require the use of the Monte Carlo Method for accurate outcomes.
  1. Why the name of this method is The Monte Carlo Method?
  2. Monte Carlo Method got this name after the city named “Monaco” in Europe. Monaco is a town mostly known for casinos.
  3. How many probability distributions are there involved in The Monte Carlo Method?
  4. There are six types of distribution probabilities.
  • Normal Distribution
  • Lognormal distribution
  • Discrete distribution
  • Triangular distribution
  • Uniform distribution
  • Pert distribution
  1. Explain the Monte Carlo Method with Palisade @risk.
  2. Users tend to apply the Monte Carlo method in the spreadsheet to analyze everyday work. The dominating tool to analyze spreadsheets is Microsoft Excel, and with palisade @risk, it enhances the accuracy of the Monte Carlo method in excel. From accuracy in calculations to easy to use chance has a significant market value. Analyzing uncertainties and risk factors of large projects is the logical application Monte Carlo method, the introduction of Microsoft project led it.
  3. What is the add-in that makes the Monte Carlo method calculate more outcomes of more accuracy?
  4. The hypercube sampling is added into the Monte Carlo method that makes it a result of more accuracy.


From the Second World War to today, the Monte Carlo method never had to look back. This method sees the exact possibilities of values that are not possible with other methods. Not only in mathematics, the monte Carlo method has it’s used in a computer language like python also.

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