Geometric Probability
Big Idea: Geometric probability uses lengths, areas, or volumes to calculate the likelihood of an event, rather than counting discrete outcomes. It extends classical probability into the continuous realm.
Formal Definition
If a point is chosen at random in a region $S$ (the sample space), and the favorable outcome corresponds to the point landing in a sub-region $A$, then:
$$P(A) = \frac{\text{measure of region } A}{\text{measure of region } S}$$
where measure can refer to:
- Length (1D problems): $P(A) = \dfrac{\text{length of } A}{\text{length of } S}$
- Area (2D problems): $P(A) = \dfrac{\text{area of } A}{\text{area of } S}$
- Volume (3D problems): $P(A) = \dfrac{\text{volume of } A}{\text{volume of } S}$