Geometric Probability
The probability of landing in a region = favorable area ÷ total area. Geometry meets probability!
Introduction
Past Knowledge
All area formulas (11.1–11.2.4). Basic probability concepts.
Today's Goal
Use area ratios to calculate geometric probability.
Future Success
Statistics, Monte Carlo simulations, game design.
Key Concepts
Geometric Probability Formula
Two Types
- 1D (length): Probability of landing on a segment = segment length / total length
- 2D (area): Probability of landing in a region = region area / total area
Worked Examples
Dart Board
Circular dartboard r = 10. Bullseye r = 2. P(bullseye)?
P = 4% or 1/25
Circle in a Square
Circle inscribed in a 10×10 square. Random point in square lands in circle?
Circle r = 5.
P ≈ 78.5%
Annulus (Ring)
Outer circle r = 8, inner circle r = 5. P(landing in the ring)?
P ≈ 60.9%
Common Pitfalls
Wrong Denominator
The denominator is always the TOTAL area, not the favorable area. P ranges from 0 to 1.
Mixing 1D and 2D
If the question is about landing on a line segment, use length ratio. If landing in a region, use area ratio.
Real-Life Applications
Carnival Games
Ring-toss and wheel-spin games are designed using geometric probability to set the odds in the operator's favor.
Monte Carlo Method
Scientists estimate π by randomly throwing points into a square and counting how many land in the inscribed circle. P ≈ π/4.
Practice Quiz
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