Geometric probability finds the chance of an event by comparing measurements of regions instead of counting separate outcomes. It is useful when there are infinitely many possible outcomes, such as where a dart lands on a board or what time a person arrives. The central idea is that probability equals the favorable size divided by the total possible size.
That size might be a length, an area, or a volume depending on the situation.
In a dartboard model, the favorable region might be a circle, ring, or sector, so area formulas are used to compute probability. In a meeting-time model, the favorable region might be an interval on a timeline, so lengths of time are compared. More advanced problems can use coordinate planes, where each point represents a pair of outcomes such as two arrival times.
The same rule applies in every case: measure the part that works, measure the whole sample space, then form a ratio.
Key Facts
- Geometric probability = favorable measure / total measure.
- For length models, P(event) = favorable length / total length.
- For area models, P(event) = favorable area / total area.
- For volume models, P(event) = favorable volume / total volume.
- Circle area formula: A = πr^2.
- For equally likely points in a region, probability depends on size of the region, not on its shape alone.
Vocabulary
- Geometric probability
- A method of finding probability by comparing lengths, areas, or volumes of favorable outcomes to the total possible region.
- Sample space
- The complete set or region of all possible outcomes in a probability experiment.
- Favorable region
- The part of the sample space that satisfies the condition described by the event.
- Uniform distribution
- A situation where every equal-sized part of the sample space has the same chance of containing the outcome.
- Area model
- A probability model in which outcomes are represented by points in a two-dimensional region.
Common Mistakes to Avoid
- Using diameter instead of radius in A = πr^2 is wrong because the radius is half the diameter and area depends on the square of the radius.
- Comparing a length to an area is wrong because the favorable and total measures must use the same type of measurement.
- Forgetting to subtract inner area from outer area for a ring is wrong because a ring is not the full outer circle.
- Assuming every drawing is uniform is wrong because geometric probability requires equal-sized lengths, areas, or volumes to be equally likely.
Practice Questions
- 1 A dart lands uniformly at random on a circular board of radius 10 cm. What is the probability that it lands within 4 cm of the center?
- 2 A student arrives randomly between 3:00 and 3:30. What is the probability that the student arrives between 3:10 and 3:18?
- 3 Two friends each arrive randomly and independently between 1:00 and 2:00. Explain how a square coordinate diagram can represent the sample space and how the region where they meet within 10 minutes would be shown.