Editing Monte Carlo Methods (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Monte Carlo methods are understood through '''Simulation''' and '''Estimation'''.

'''1. The "Darts" Method (Integration)''':
How do you find the area of a "Blob" if you don't have a formula?
* You draw a "Square" around the blob.
* You "Throw 1,000,000 random darts" at the square.
* You count how many darts "Hit the blob" and how many "Hit the square."
* The "Ratio" of hits gives you the "Area" with incredible accuracy.

'''2. The "What-If" Machine (Forecasting)''':
In finance, we don't know "The" price of a stock next year.
* We know the stock has a "Mean" and a "Volatility."
* We run 10,000 "Possible Futures" (Paths) for the stock.
* Some paths go to Zero; some go to the Moon; most stay in the middle.
* By looking at the "Distribution" of these 10,000 futures, we can say: "There is a 5% chance of a crash."

'''3. The "Infinite Monkey" Principle''':
Monte Carlo is about "Trying everything."
* It doesn't need to "Understand" the physics or the logic.
* It just "Plays the game" over and over.
* This is why it works for "Complex systems" (like Weather or Galaxy formation) where the "Equations" are too hard to solve.

'''The 'Solitaire' Origin'''': Stan Ulam was bored. He wondered: "What are the odds of winning this game of Solitaire?" He tried to use "Combinatorics" (math), but it was too hard. He realized: "I can just play 100 games, count the wins, and that's my answer." He told John von Neumann, and the Monte Carlo method was born.
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