Concept of Geometric Probability

According to Eric Weisstein's World of Mathematics, we have the following definitions.

Geometric Probability: The study of the probabilities involved in geometric problems; e.g., the distributions of length, area, volume, etc. for geometric objects under stated conditions. Sample Space: Informally, the sample space for a given set of events is the set of all possible values the events may assume.

Certain events may be considered "successful". To this end we give the following definition.

Event Space: The subset of the sample space consisting of events that represent a successful outcome.

We introduce now two general methods for finding the probability of a success:

Theoretical Probability: The predicted probability of a successful outcome based on a mathematical model. Empirical Probability: The actual probability of a successful outcome when a simulation is run.

Now suppose a dart is thrown at a 9m square target (see the following diagram below) in such a way that the dart is equally likely to hit one point of the square as any other. The entire square represents the sample space. Every dart thrown in the center 3m square (shaded green in the diagram) will be considered a successful outcome; this green region represents the event space. To calculate the probability of hitting the green square, we compare its area to that of the entire 9m square: thus we have our theoretical computation of the probability of a successful outcome as 9/81 = 1/9.

What is the probability that the dart will hit a specific point in the dartboard, such as the exact center? If there were a finite number of points in the board, say one billion, the odds of hitting the bullseye would be one out of a billion. Because there are infinitely many points on the dartboard, the probability is actually zero: this does not mean that it is impossible for the dart to land on a single point, as of course the dart must hit some point, and the probability of hitting that point is 0. This brings us to the following two definitions.

Discrete case: A sample space that has a finite number of outcomes. Continuous case: A sample space that has an infinite number of outcomes.

Now consider the following example. Bill and Yolanda want to meet at Sweet Things for an after dinner treat. They agree to arrive between between 8:00 PM and 8:30 PM; they also agree that the first one to arrive shall purchase two cones and wait for the other person to arrive. If, however, the second person does not arrive within 12 minutes, then the first person will start to eat that person's cone while continuing to wait for the friend. If they arrive at the same time, each will buy his own cone. What is the probability that each person eats no more than one ice-cream cone?

The sample space can be represented on the coordinate plane by a 30x30 square. Let x and y denote the number of minutes after 8:00 PM that Bill and Yolanda arrive at Sweet Things, respectively. If they arrive within 12 minutes of each other, then each will have to eat only one ice-cream cone. The event space (where the event is both arriving within 12 minutes of each other) is then equal to the sample space under the extra constraint

x - 12 < y < x + 12.

Thus, all possible successful events are indicated by the shaded region below; the probability of a success is the ratio of the shaded region's area to that of the square.

Now the area of the shaded region is the total area of the square, 30² = 900, minus the area of the two white triangles, or 2.(1/2).18.18=324. Thus, the probability of a successful outcome is (900 - 324) / 900 = 0.64.