Monte Carlo simulation is a way to solve problems by using random trials instead of exact step-by-step calculation. In this school project, students estimate π by dropping random points into a square that contains a quarter circle. The method is useful because it connects geometry, probability, coding, and data analysis in one visual experiment.
It also shows why more data usually leads to a better estimate, but not always in a perfectly smooth way.
Key Facts
- Use random points (x, y) where 0 <= x <= 1 and 0 <= y <= 1.
- A point is inside the quarter circle if x^2 + y^2 <= 1.
- Area of unit square = 1.
- Area of quarter circle with radius 1 = π/4.
- inside/total ≈ π/4, so π ≈ 4 * (inside/total).
- Increasing the number of trials usually reduces random error, but results still fluctuate.
Vocabulary
- Monte Carlo simulation
- A method that uses many random trials to estimate a result or model a process.
- Random point
- A point whose coordinates are chosen unpredictably within a specified range.
- Trial
- One repeated test in a simulation, such as generating one random point.
- Convergence
- The process of an estimate getting closer to a stable value as more trials are added.
- Unit circle
- A circle with radius 1, usually centered at the origin on a coordinate plane.
Common Mistakes to Avoid
- Using x + y <= 1 instead of x^2 + y^2 <= 1 is wrong because the circle rule comes from the distance formula, not from adding coordinates.
- Forgetting to multiply by 4 is wrong because the random points estimate the area of only one quarter of the circle.
- Expecting every run to give the same π estimate is wrong because random points create natural variation from run to run.
- Using too few trials is misleading because small samples can produce estimates that are far from π due to random clustering.
Practice Questions
- 1 A simulation generates 1000 random points in the unit square, and 782 points land inside the quarter circle. Estimate π using π = 4 * (inside/total).
- 2 A student wants an estimate of π equal to 3.16 after 5000 trials. How many points must have landed inside the quarter circle?
- 3 Two students run the same Monte Carlo π project. One uses 200 points and the other uses 20000 points. Explain which estimate is more likely to be closer to π and why, while also explaining why neither result is guaranteed to be exact.