Monte Carlo & Buffon's Needle

[Pt] Monte Carlo & Buffon's Needle demonstrates how randomness can solve deterministic geometric problems. In Sphere in Cube (3D) mode, random points (x, y, z) rain into a cube of side 2R; because the inscribed sphere occupies exactly π/6 ≈ 52.36% of the cube's volume, multiplying the fraction of gold inside hits by 6 estimates π. Students can also switch to the classic 2D Circle in Square ratio (π/4) or Buffon's Needles, where sticks of length L dropped onto floorboards spaced D apart cross a line with probability 2L/(πD). Dragging the overhead emitter or firing 500-sample bursts drives the live convergence curve toward 3.14159 by the Law of Large Numbers.

• [Pt] Watch the early samples in Sphere in Cube (3D) mode fluctuate widely on the convergence chart, then click Drop 500 Burst several times to see the curve settle near 3.1416.
• [Pt] Drag or click the glowing gold emitter ring above the cube to rain concentrated bursts of darts into the volume.
• [Pt] Switch Simulation mode to Circle in Square (2D) and compare the hit percentage (~78.5% = π/4) with the 3D sphere hit percentage (~52.4% = π/6).
• [Pt] Select Buffon's Needles mode to drop 3D sticks onto parallel ruled lines and see crossing needles light up gold.
• [Pt] Use Clear samples and run three separate trials of N = 500 to observe statistical variance in Monte Carlo estimation.

1. 1[Pt] Derive the geometric probability ratios V_sphere / V_cube = π/6 and A_circle / A_square = π/4.
2. 2[Pt] Explain how the Law of Large Numbers guarantees that a Monte Carlo sample proportion K/N converges to the true probability as N grows.
3. 3[Pt] Describe how Buffon's Needle connects geometric probability of random angles and positions to π.
4. 4[Pt] Analyze why Monte Carlo estimation error decreases proportionally to 1/√N.

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