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Geometric Probability
Introduction to geometric probability and its applications in math contests.
Authors: Jaden Park, Alexander Ren
Geometric probability involves the study of outcomes in a geometric setting, often dealing with measurements such as lengths, areas, and volumes. This area of probability allows us to solve problems where traditional discrete or continuous probability methods might not apply directly.
Geometric probability can be defined when an experiment's outcomes are points in a geometric region. The probability of an event is the ratio of the measure of the event's region to the measure of the entire space.
For instance, if we consider a line segment of length and an event that a point chosen at random on this line segment lies in a particular subsegment of length , the probability of this event is:
Here are some examples to illustrate the concept.
There is a circular dartboard with a diameter of cm. Within the dartboard is a smaller circle (the bullseye) with a diameter of cm. If a player throws a dart at the board randomly, what is the probability that the dart hits the bullseye?
The area of the entire dartboard is:
The area of the bullseye is:
Thus, the probability of hitting the bullseye is:
Consider a rectangle with length cm and width cm. A point is chosen at random inside this rectangle. What is the probability that the point is closer to the rectangle's perimeter than to its center?
The rectangle has a length of cm and a width of cm. Its center is therefore at the point , assuming one corner of the rectangle is at the origin .
For a point to be closer to the perimeter than to the center, it must lie outside the central rectangle that is equidistant from the perimeter and the center. This central rectangle's dimensions can be determined by the maximum distance a point can be from the center while still being closer to it than to any side of the larger rectangle.
The maximum distance from the center to the perimeter along the length is half of the smaller side, i.e., cm. Therefore, the central rectangle that a point can lie in and still be closer to the center than the perimeter will have its length and width halved, resulting in dimensions of cm by cm.
Now, we can calculate the areas:
The area outside this smaller rectangle but inside the larger one is where a point will be closer to the perimeter. This area is the difference between the two rectangles' areas:
Therefore, the probability that a randomly chosen point inside the larger rectangle is closer to the perimeter than to the center is the ratio of the area where this condition is true to the total area of the rectangle:
Hence, the probability is or .
Here are some more problems.
Two points are chosen randomly and uniformly along a stick of length The stick is cut at those points to form pieces. What is the probability that these pieces can form a triangle?
Two friends who take metro to their jobs from the same station arrive to the station uniformly randomly between and in the morning. They are willing to wait for one another for minutes, after which they take a train whether together or alone. What is the probability of their meeting at the station?
(1992 Putnam A6)
Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?