Geometric Probability
Calculate probabilities as ratios of lengths, areas, or volumes. Apply to dart board problems, length problems on a segment, and area problems with circles and rectangles inscribed within larger figures.
A lesson of Geometry, in Area and Volume, about 10 minutes.
By the end of this lesson, you will be able to compute geometric probabilities using length, area, and volume ratios, interpret "random point" problems geometrically, and apply these concepts to real-world scenarios.
The Big Idea
Geometric probability uses length, area, or volume ratios instead of counting outcomes. When a point is chosen "at random" in a region, the probability of landing in a sub-region is:
$$P(\text{event}) = \frac{\text{measure of favorable region}}{\text{measure of total region}}$$
"Measure" means length for 1D, area for 2D, volume for 3D.
Length-Based Probability
Probability of landing on a segment:
$$P = \frac{\text{length of favorable segment}}{\text{length of total segment}}$$
Area-Based Probability
Probability of landing in a region:
$$P = \frac{\text{area of favorable region}}{\text{area of total region}}$$
Example 1 — Segment (1D)
A stick of length 20 cm is broken at a random point. What is the probability the break is within 6 cm of one end?
Favorable: within 6 cm of left end (0 to 6) or right end (14 to 20) → total favorable length = 12 cm.
$$P = \frac{12}{20} = 0.6 = 60%$$
Example 2 — Circle in a Square
A square has side 12. A circle of radius 4 is inscribed inside (centered at the square's center). A random point is chosen inside the square. What is the probability it lands inside the circle?
$$P = \frac{\pi(4)^2}{12^2} = \frac{16\pi}{144} = \frac{\pi}{9} \approx 0.349 \approx 34.9%$$
Example 3 — Dartboard (Concentric Circles)
A circular dartboard has radius 15 inches. The bullseye is a circle of radius 3 inches. What fraction of throws (randomly aimed) hits the bullseye?
$$P = \frac{\pi(3)^2}{\pi(15)^2} = \frac{9}{225} = \frac{1}{25} = 4%$$
Example 4 — Shaded Region
A rectangle is 10 × 8. A triangle with base 10 and height 6 is inside the rectangle. A random point in the rectangle is chosen. What is the probability it lands inside the triangle?
$$P = \frac{\frac{1}{2}(10)(6)}{10 \times 8} = \frac{30}{80} = \frac{3}{8} = 37.5%$$
Example 5 — Arc Length (1D on Circle)
A point is chosen randomly on the circumference of a circle. A chord subtends an arc of 72°. What is the probability the point lands on that arc?
$$P = \frac{72°}{360°} = \frac{1}{5} = 20%$$
Buffon's Needle (Famous Application)
Buffon's Needle Problem: drop a needle of length $\ell$ at random onto a floor with parallel lines spaced $d$ apart ($\ell \leq d$). The probability it crosses a line is: