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# Monte Carlo Method

Monte Carlo Simulation, also known as the Monte Carlo Method or a multiple probability simulation, is a mathematical technique, which is used to estimate the possible outcomes of an uncertain event. The Monte Carlo method is named after the city in Monaco, famous for its casinos and gambling. The term was coined together because the principles of randomness involved in this method mimic a game of roulette.

The method has become hugely popular in many real-life scenarios, such as stock prices, sales forecasting, project management, artificial intelligence, and pricing.

They also provide several advantages over predictive models with fixed inputs, the main ability being able to conduct sensitivity analysis or calculate the correlation of inputs. Sensitivity analysis allows decision-makers to see the impact of individual inputs on a given outcome and correlation allows them to understand relationships between any input variables.

So, why is it so popular in our industry?

1. Probabilistic Results. Results show not only what could happen, but how likely each outcome is.

2. Graphical Results. Because of the data, a Monte Carlo simulation generates, it’s easy to create graphs of different outcomes and their chances of occurrence. This is important for communicating findings to other stakeholders.

3. Sensitivity Analysis. With just a few cases, the deterministic analysis makes it difficult to see which variables impact the outcome the most. In the Monte Carlo simulation, it’s easy to see which inputs had the biggest effect on bottom-line results.

4. Scenario Analysis: In deterministic models, it’s very difficult to model different combinations of values for different inputs to see the effects of truly different scenarios. Using Monte Carlo simulation, analysts can see exactly which inputs had which values together when certain outcomes occurred. This is invaluable for pursuing further analysis.

5. Correlation of Inputs. In Monte Carlo simulation, it’s possible to model interdependent relationships between input variables. It’s important for accuracy to represent how when some factors go up, others go up or down accordingly.

Let us take an example to understand:

Imagine a marble-dropping device that moves randomly above two containers. One container is round and the other square. The round container is also slightly larger in size. Both these containers are placed on some scale that can help measure the number of marbles that are within the containers.

Once the simulation starts the device randomly drops balls into both these containers in no specific order. After the simulation runs for some time and stops, we go ahead and note the count of balls in each container with the help of the scale. Once the readings are obtained, we divide the count from the round container by that of the square and we can see that the obtained result can be approximated to 3.14, which is the value of pi.

This simulation proves that the probability of a ball falling into a container is directly proportional to the cross-section area of the container. If the square container has a side of dimension ‘a’ and the radius of the circular container is also ‘a’, then the area of the square becomes ‘a*a’ and the area of the circular container becomes ‘a pi*a*a’. On dividing them, the approximation totals to pi, hence proved by the Monte Carlo simulation.

The random samples give us an idea about whether they fall inside or outside of an area, and by taking enough samples we get an idea of how big the area is.

Now, how is this mathematical experiment relevant in the real world?

Imagine a scenario where you want to calculate the average height of all the people in the world. To ideally obtain this, we would need to take the height of each individual person across the plane which is not a feasible task!

So, we stick to taking the height of a small group of people who represent various varieties. To ensure there is no bias in the way people are selected we opt to randomly select people across the world for this experiment. With this method of deliberately employing randomness and with the law of large numbers, with enough samples, we can achieve our desired result.

Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does these hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, the Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen.

The idea of randomness to predict the future is something that is truly inspiring, but better even with the fact that it is all backed up mathematically making our decisions more trustworthy!