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Monte Carlo Simulation — Understanding Probability Through Randomness - Leejohnston - 11-17-2025
Monte Carlo Simulation — Understanding Probability Through Randomness
How Computers Use Randomness to Solve Problems We Can’t Solve Directly
Monte Carlo methods are one of the most powerful ideas in modern mathematics, science, and computing.
They allow us to understand complex systems by simulating them thousands — or millions — of times using randomness.
From estimating π, to predicting weather, to valuing financial risks, to modelling nuclear reactions…
Monte Carlo simulation is everywhere.
This thread introduces the core idea in a clean, accessible way.
1. What Is a Monte Carlo Simulation?
A Monte Carlo simulation is a method where we:
• define a problem
• add randomness
• run the scenario many times
• analyse the results
Instead of solving a problem exactly (often impossible),
we approximate the answer by simulating reality over and over.
Examples:
• simulating coin flips
• modelling population growth
• predicting climate behaviour
• estimating the probability of rare events
• physics simulations for particle interactions
It’s “mathematical experimentation.”
2. Why Use Monte Carlo Methods?
Some problems are:
• too complex for exact formulas
• too chaotic
• too uncertain
• too high-dimensional
• influenced by random variables
Monte Carlo lets us explore those systems anyway.
Advantages:
• works for almost any problem
• handles uncertainty naturally
• increasingly accurate with more simulations
• easy to implement with computers
Modern AI and physics simulations rely heavily on this.
3. A Simple Example — Estimating π
The classic beginner Monte Carlo task:
1. Draw a square.
2. Inscribe a circle inside it.
3. Throw random points into the square.
4. Count how many land inside the circle.
Because the area of a circle is πr² and the area of the square is (2r)²,
the ratio inside/total approximates π/4.
So π ≈ 4 × (points inside circle ÷ total points).
The more points → the more accurate.
4. Real-World Uses of Monte Carlo
Physics & Engineering
• nuclear simulations
• radiation modelling
• fluid dynamics
• astrophysical systems
• reactor design
Finance & Economics
• stock price modelling
• risk estimation (Value at Risk)
• portfolio uncertainty
• option pricing
Computer Science & AI
• reinforcement learning
• game AI (e.g., AlphaGo)
• pathfinding under uncertainty
Medicine & Biology
• disease spread modelling
• genetic population simulations
• medical imaging reconstruction
Everywhere: When the real world is uncertain, Monte Carlo thrives.
5. How a Monte Carlo Simulation Works (Step by Step)
Step 1 — Define the model
What system are we studying? What variables matter?
Step 2 — Add randomness
Assign probability distributions to uncertain elements.
Step 3 — Run many simulations
Examples: 1,000, 10,000, or millions of runs.
Step 4 — Collect results
Record outcomes: averages, extremes, probabilities.
Step 5 — Analyse patterns
Look for trends, convergence, or risk levels.
Monte Carlo transforms uncertainty into measurable behaviour.
6. Why Monte Carlo Always Works (Eventually)
The magic behind Monte Carlo is the:
Law of Large Numbers
→ with enough trials, the average result converges to the true probability.
So even if each simulation is chaotic…
the combined results reveal order inside randomness.
Written by Leejohnston & Liora
The Lumin Archive — Statistics & Probability Division