Geometric Probability
Turn continuous uniform probability questions into areas.
What is it?
When variables are uniform on an interval or region, probabilities equal the favorable area (or volume) divided by the total.
Why is it useful?
Meeting problems, broken sticks and ratio questions reduce to drawing a region in the unit square.
When should you think of it?
Use it when two or three independent uniform variables appear and the event is an inequality between them.
Think of this method when…
- Independent variables, or "at random" on a segment
- An inequality like or
- Meeting times, broken sticks, random points
- Two or three variables, so a picture is drawable
Core idea
The joint density is flat, so probability = area of the favorable region.
Step-by-step playbook
- 1Normalize lengths so the variables live in .
- 2Draw the unit square (or cube).
- 3Shade the region where the event holds.
- 4Compute its area using triangles and rectangles.
Worked example
Worked example
Two people arrive independently and uniformly in a one-hour window and each waits 30 minutes. What is the probability they meet?
- 1
With they meet when .
- 2
The complement is two corner triangles, each of area .
- Answer:
Common mistakes
- Forgetting the region is clipped by the square's edges.
- Using it for non-uniform variables without the density.
Would you use this method?
. What is ?
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