Geometric probability (area-based)
Geometric (Area-Based) Probability
Geometric (Area-Based) Probability is a topic in Probability & Statistics in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.13(B), which requires students to determine probabilities based on area to solve contextual problems.
Geometric probability finds the chance a random point lands in a region as the ratio of the favorable area to the total area.
Theory
where the measure is length on a segment, area in a region, or volume in a solid.
The area (or length) ratio:
How to find a geometric probability
- Identify the total region and the favorable region.
- Measure each (length, area, or volume).
- Divide favorable by total.
- Keep the measures the same type and simplify.
Divide the circle's area by the square's area.
| \(P\) | \(=\) | \(\dfrac{\pi r^2}{(2r)^2}\) |
| \(=\) | \(\dfrac{\pi r^2}{4r^2}=\dfrac{\pi}{4}\approx0.785\) |
Use the ratio of lengths.
| \(P\) | \(=\) | \(\dfrac{3}{10}=0.3\) |
Divide the shaded area by the total area.
| \(P\) | \(=\) | \(\dfrac{4\times4}{10\times10}\) |
| \(=\) | \(\dfrac{16}{100}=0.16\) |
Divide the bullseye area by the whole target area.
| \(P\) | \(=\) | \(\dfrac{\pi(2)^2}{\pi(6)^2}\) |
| \(=\) | \(\dfrac{4}{36}=\dfrac19\) |
Common pitfalls
Frequently asked questions
What is geometric probability?
The probability a random point lands in a region, found as the ratio of the favorable measure to the total measure.
How do you compute it?
Divide the favorable area (or length or volume) by the total.
What assumption does it make?
That the point is uniformly random — equally likely anywhere in the region.
What is the probability of a dart in an inscribed circle?
\(\dfrac{\pi r^2}{(2r)^2}=\dfrac{\pi}{4}\approx0.785\).