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.
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.
Answer: 2/5
Practice problems
1. Point chosen on segment 0 to 20. P(point > 15) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Point chosen on segment 0 to 20. P(point > 15) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Favorable length: 20 - 15 = 5.
- P = 5/20 = 1/4.
- Check the result by substituting or estimating: the response should match 1/4 and make sense in the original problem.
Answer: 1/4
2. Point chosen on segment 0 to 12. P(between 4 and 10) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Point chosen on segment 0 to 12. P(between 4 and 10) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Favorable: 10 - 4 = 6.
- P = 6/12 = 1/2.
- Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.
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
- Warm-up: First identify exactly what the question is asking: 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.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Areas: 25 / 100 = 1/4.
- Check the result by substituting or estimating: the response should match 1/4 and make sense in the original problem.
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
- Core Practice: First identify exactly what the question is asking: Random point in a 20-by-20 square. A 4-by-4 sub-square is inside. P(inside it) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- 16 / 400 = 4/100.
- Check the result by substituting or estimating: the response should match 4/100 and make sense in the original problem.
Answer: 4/100
5. Random point on segment 0 to 24. P(less than 6) as a fraction.
Show solution
- Core Practice: First identify exactly what the question is asking: Random point on segment 0 to 24. P(less than 6) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- 6 / 24 = 1/4.
- Check the result by substituting or estimating: the response should match 1/4 and make sense in the original problem.
Answer: 1/4
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