Bear With Me: Monte Carlo and Me and Perplexity

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Monte Carlo methods originated in the mid-20th century, although early concepts date to the 18th century. The pivotal modern development happened during World War II with Stanislaw Ulam, who, while working at Los Alamos National Laboratory on nuclear weapons research, conceived random sampling as a way to solve complex neutron diffusion problems that defied standard mathematical analysis.wikipedia+6

Early Mathematical Roots

The initial concept can be traced to experiments like Buffon’s needle in the 18th century, where French scientist Georges Louis LeClerc (Comte de Buffon) used repeated random trials to estimate mathematical values, notably Pi.pmc.ncbi.nlm.nih+1
In the 19th century, simulation and randomization methods continued to develop in statistics and probability studies, such as smoothing mortality tables and generating distributions with dice and cards.pmc.ncbi.nlm.nih

World War II and the Manhattan Project

The method’s breakthrough came in the 1940s at Los Alamos. Stanislaw Ulam, recovering from illness and playing solitaire, realized some problems—like predicting the likelihood of winning the game or neutron movement in reactors—could be addressed by simulating many random outcomes and observing their frequencies.oakleyj.github
Ulam shared his idea with John von Neumann, who recognized its power and helped formalize the computational implementation. The approach was first coded for the ENIAC electronic computer in 1948.lanl+1
The term “Monte Carlo” was coined by Nicholas Metropolis, inspired by both the randomness of the method and the famed Monte Carlo Casino, symbolizing luck and probabilistic games.investopedia+1

Key Contributors

Stanislaw Ulam: Conceived and refined the central concept.oakleyj.github
John von Neumann: Helped formalize and implement the numerical technique.aws.amazon+1
Nicholas Metropolis: Named the method and contributed to coding and applications.metaplane+1

Monte Carlo methods revolutionized simulation and modeling, becoming foundational in physics, finance, engineering, and data science.investopedia+2

What Are Monte Carlo Simulations?

Monte Carlo simulations are mathematical techniques that use random sampling from defined probability distributions to create a wide range of possible scenarios for a given problem. Instead of relying on single-point predictions, the process repeats thousands—even millions—of iterations, each drawing random values for input variables. This results in a comprehensive picture of potential outcomes and their likelihoods.investopedia+3

How Do They Work?

The basic workflow involves:

Defining uncertain variables (such as market returns, costs, project delays).mastt+2
Assigning each variable a probability distribution that reflects its possible range (e.g., normal, triangular, uniform).mastt
Running thousands of iterations, each randomly sampling values from these distributions.lumivero+1
Aggregating all results to produce a probability distribution of outcomes.lumivero+1

For example, in finance, if one wants to forecast retirement portfolio sustainability, a Monte Carlo Simulation would vary market returns, inflation rates, interest rates, and more, then review the range and probability of possible outcomes.emoneyadvisor+2

Real-World Applications

In finance, Monte Carlo methods help analyze:

Portfolio return variability and risk
Retirement income sustainability projectionsanalytica+1
Stock option pricing by simulating asset price paths and payoffsinvestopedia+1
Asset-liability modeling in pensions and insurance
Risk assessment, including “Value at Risk” and stress testinginteractivebrokers+1

Beyond finance, these methods are widely used for:

Project cost, schedule, and risk modeling in engineeringlumivero+1
Reliability and efficiency forecasts in manufacturinglumivero
Estimating solutions to complex mathematical problems, such as approximating Pi or infinite seriesmetaplane

A Simple Example: Predicting Stock Price

First, I’m going to outline the problem we are going to use Monte Carlo techniques to address, then write a Perplexity Prompt to actually use Monte Carlo to solve, and lastly, the results provided by Monte Carlo. At this point, you may be asking yourself Why? It falls in the “because I can class” and also to point out some of the less evident things Perplexity can do for you, if you are so inclined.

I have been tracking and buying a significant position in a stock with the ticker TRIN. It is a BDC company with a current 13.1% simple annualized return, pays dividends quarterly, at $0.51 per share. Recently, the stock dropped 4.5% in a single day, which I viewed as a buying opportunity. I had been buying TRIN at over $15.00, and accordingly, it was now in the $14.40 vicinity. I bought 1000 shares at 14.005.

Let’s assume TRIN’s price fits a triangle distribution. The triangular distribution is a continuous probability distribution shaped like a triangle, defined by three parameters: the minimum value aa, the maximum value bb, and the mode (or most likely value) cc, where a<ba<b and a≤c≤ba≤c≤b. It’s widely used when limited data is available, as it allows for the estimation of these three values, making it popular in project management, risk analysis, and Monte Carlo simulation work.wikipedia+4

You can generally get C-level executives to estimate next year’s sales as worst case, best case, and most likely after pounding heads for a while. Same thing for other estimates, which is the power of the triangle distribution. For TRIN, let’s assume the lowest price to expect is $14.00, the highest 18, and the most likely 16. We have everything we need to run a Monte Carlo simulation except the number of iterations. Let’s arbitrarily set this at 100.

Prompt: Run a Monte Carlo using the following: a triangular distribution with lowest value 14, highest 18, and most likely 16, using 100 iterations. Place the results in a table along with the probabilities of each histogram bin you provide.

Here is the extended table including the probability values for each bin in the Monte Carlo simulation from a triangular distribution with minimum 14, mode 16, and maximum 18, based on the PDF integration over each bin’s interval:

Bin Range	Count	PDF Value (Height)	Probability (Area of bin)
14.0–14.8	10	0.125	0.100
14.8–15.6	18	0.3125	0.225
15.6–16.4	38	0.5	0.320
16.4–17.2	22	0.3125	0.225
17.2–18.0	12	0.125	0.130

Explanation:

Count: Number of samples in the bin from the simulation.
PDF Value: The height of the triangular probability density function at the midpoint of the bin.
Probability: The integral (area) of the PDF over the bin range, representing the probability that a random variable falls within that interval.

These probability values sum approximately to 1 (due to rounding), confirming that the bins collectively cover the entire probability space for this distribution. This table reflects how the triangular distribution assigns higher probabilities near the mode and lower toward the minimum and maximum values, aligning with the Monte Carlo simulation output.statology+2

And that, kind folks, is how you make Perplexity do weird statistical stuff for you!