Geometric Probability
A free Geometry lesson from the “Probability” unit, with a worked example and practice problems including step-by-step solutions.
Geometric probability uses continuous measures (length, area, volume) instead of counts. The probability of a random point landing in a favorable region is (measure of favorable region) / (measure of total region). For a number line: favorable length / total length. For a 2D figure: favorable area / total area. In Probability, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.
What you'll learn
- Compute geometric probability as a ratio of favorable measure (length or area) to total measure
- Apply length-based geometric probability on a segment or number line
- Apply area-based geometric probability inside a region
Worked example
Problem. A point is chosen at random on the segment from 0 to 10. Find P(point lands between 3 and 7) as a fraction.
- Favorable length: 7 - 3 = 4.
- Total length: 10 - 0 = 10.
- Probability = 4 / 10 = 2/5.
- Connect the result back to Geometric Probability so the geometric relationship is explicit.
Answer: 2/5
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Point chosen on segment 0 to 20. P(point > 15) as a fraction.
Show solution
- Favorable length: 20 - 15 = 5.
- P = 5/20 = 1/4.
Answer: 1/4
2. Point chosen on segment 0 to 12. P(between 4 and 10) as a fraction.
Show solution
- Favorable: 10 - 4 = 6.
- P = 6/12 = 1/2.
Answer: 1/2
3. Random point in a 10-by-10 square. A 5-by-5 sub-square sits in one corner. P(inside the sub-square) as a fraction.
Show solution
- Areas: 25 / 100 = 1/4.
Answer: 1/4
4. Random point in a 20-by-20 square. A 4-by-4 sub-square is inside. P(inside it) as a fraction.
Show solution
- 16 / 400 = 4/100.
Answer: 4/100
5. Random point on segment 0 to 24. P(less than 6) as a fraction.
Show solution
- 6 / 24 = 1/4.
Answer: 1/4
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