🎲 Geometric Probability

Geometric probability problems combine geometry with probability, often involving random points, lines, or regions. The key is finding the ratio of favorable outcomes to total outcomes.

🎯 Key Concepts

Basic Principle

Fundamental: Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$

In geometric probability:

  • Favorable outcomes = favorable geometric measure (length, area, volume)
  • Total outcomes = total geometric measure (length, area, volume)

Common Problem Types

  1. Random points on line segments
  2. Random chords in circles
  3. Random points in regions
  4. Random lines through figures

Probability Formulas

  • Length probability: $P = \frac{\text{Favorable length}}{\text{Total length}}$
  • Area probability: $P = \frac{\text{Favorable area}}{\text{Total area}}$
  • Volume probability: $P = \frac{\text{Favorable volume}}{\text{Total volume}}$

🔍 Micro-Examples

Random Point on Segment

Point chosen randomly on segment of length 10. Probability it’s within 3 units of one end:

  • Favorable length: 6 (3 from each end)
  • Total length: 10
  • Probability: $\frac{6}{10} = \frac{3}{5}$

Random Chord in Circle

Chord chosen randomly in circle of radius 5. Probability it’s longer than radius:

  • Favorable: Chords longer than 5
  • Total: All possible chords
  • Need to calculate areas or use geometric properties

Random Point in Region

Point chosen randomly in square of side 4. Probability it’s within 1 unit of center:

  • Favorable area: Circle of radius 1 = $\pi$
  • Total area: Square = 16
  • Probability: $\frac{\pi}{16}$

⚠️ Common Traps

Pitfall: Wrong geometric measure

  • Fix: Use length for 1D, area for 2D, volume for 3D

Pitfall: Forgetting to check if point is in region

  • Fix: Make sure random point satisfies given conditions

Pitfall: Wrong probability formula

  • Fix: Always use $\frac{\text{Favorable}}{\text{Total}}$

Pitfall: Forgetting units

  • Fix: Make sure units match (length with length, area with area)

🎯 AMC-Style Worked Example

Problem: A point is chosen randomly inside a square with side length 6. What is the probability that the point is within 2 units of the center?

Solution: First, find the total area:

  • Square area = $6^2 = 36$

Next, find the favorable area:

  • Circle centered at square center with radius 2
  • Circle area = $\pi \cdot 2^2 = 4\pi$

Now, check if the circle fits entirely within the square:

  • Circle diameter = $2 \cdot 2 = 4$
  • Square side = 6
  • Since $4 < 6$, the circle fits entirely within the square

Therefore, the probability is: $P = \frac{4\pi}{36} = \frac{\pi}{9}$

Answer: $\frac{\pi}{9}$

🔗 Related Topics

💡 Quick Reference

Common Problem Types

  • Line segments: Use length ratios
  • Circles: Use area ratios
  • Squares/rectangles: Use area ratios
  • Triangles: Use area ratios

Probability Formulas

  • Length: $P = \frac{\text{Favorable length}}{\text{Total length}}$
  • Area: $P = \frac{\text{Favorable area}}{\text{Total area}}$
  • Volume: $P = \frac{\text{Favorable volume}}{\text{Total volume}}$

Common Values

  • Circle area: $\pi r^2$
  • Square area: $s^2$
  • Triangle area: $\frac{1}{2}bh$
  • Rectangle area: $lw$

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