Geometric Probability

Geometric probability is the calculation of probability using geometric measures. Probabilities involving continuous outcomes are represented along a number line or across a two-dimensional plane. E.g., an experiment having numerous probabilities is often shown as a geometric probability.
To calculate geometric probability, we simply need to divide the desired area by the total area.
 

The probability that a randomly selected point on a line segment falls inside a specific area of that segment is known as 1-dimensional geometric probability. This kind of probability uses length as a measure rather than counting outcomes. The favorable length—the portion of the line where the event takes place—is divided by the segment's overall length to determine the probability.

   Probability = Favorable measure/Total measure 

E.g., if a point is randomly selected on a 10-unit line and the desired segment is 3 units long, then the probability is 3/10, or 30%. 
Another example would be waiting for a bus that could arrive at any time between 10:00 and 11:00 AM. The probability that it would arrive within a 10-minute window is 10/60 = 1/6, or roughly 16.67%. This approach is helpful when outcomes are continuous and can be measured by length or distance.

The probability that a randomly chosen point within a two-dimensional region, like a rectangle or a square, falls inside a specified sub-region, like a circle or shaded area, is known as 2-dimensional geometric probability. This probability is calculated by comparing the area of the favorable region to the area of the entire region, rather than counting outcomes. The fundamental formula is

           Probability = Favourable Area/Total Area

For example, let’s take a square whose sides are four units long. It has a total area of square units that is 4 × 4 = 16 square units. There is a circle with a radius of one unit inside the square r2. The circle's area is π × 12 =   3.14 square units. The likelihood that a point will fall inside the circle if it is positioned at random within the square is

   P = 16  0.196 or 19.6%

We can even understand this with the help of a figure below:

The 3-dimensional geometric probability helps find the chance that a random point inside a 3D space lands within a specific sub-region of that 3D space. Here, the 3D space can be anything from a cube to a sphere. This probability compares volumes rather than counting discrete outcomes. The following formula is applied:

                                                   
                                                           Probability = Favourable Volume/Total Volume

For example, the volume of a cube with 4 units of side length is 4³ = 64 cubic units. Now picture a sphere inside this cube with a radius of one unit. The sphere's volume is  43r3 = 43 13   4.19 cubic units. 
Thus, the likelihood of a point chosen randomly within the cube falling inside the sphere is:

  P = Favourable Volume/Total Volume = 4.1964 / 0.0655 or 6.55 %

When spatial randomness is involved, this idea is particularly helpful in simulation modeling, engineering, and physics. We can better understand the concept with the help of a diagram:
 

Geometric probability is expressed by comparing a favorable region to the entire region using length, area, or volume. Geometric probability makes use of measurement rather than counting outcomes, as is the case with classical probability. For example, in case of a 2D object, let’s assume a circle is placed inside a square. Now when a point is chosen randomly, the probability of the point landing inside the circle is determined by dividing the area of the circle by the area of the square. In this representation, the advantageous area is prominently shaded or marked within the total space using both mathematical formulas and visual diagrams. By comparing sizes instead of counts, these visual aids help make the probability easier to understand.

Numerous real-world situations, such as risk assessment, navigation, and design, use geometric probability. It is used to predict traffic flow, optimize resource placement, analyze spatial data in geography and urban planning, and determine the probability that an event will occur within a given area. Additional uses include environmental science for estimating animal populations in a particular area, computer graphics, and quality control in manufacturing.

Traffic and Urban Planning

City planners can create safer and more effective roads, intersections, and pedestrian pathways by using geometric probability. Planners can make data-driven decisions by examining the likelihood that cars will enter specific zones, or by placing road signs within specific ranges.
For example, traffic engineers can use geometric probability to calculate the probability that a car will enter a pedestrian zone at random by comparing the pedestrian zone's area to the intersection's overall area. The probability is 25/100 = 0.25, or 25%, if the pedestrian zone takes up 25 m² of space within a 100 m² intersection.

Gaming and Sports

In sports, particularly in games where targets are defined geometrically, such as basketball, darts, or golf, geometric probability can be used to estimate shot success rates. For instance, let’s say a basketball player's shooting range creates a semicircle with a radius of two meters. If the hoop's diameter is 0.45 meters, then the coach can use the hoop's area in relation to the entire shooting zone to calculate the likelihood of a basket from that position.

Scientific Simulation and Experiments

In simulations involving space and randomness, like particle collisions, radiation spread, or molecular motion, researchers employ geometric probability.

For instance, in a laboratory experiment, researchers might simulate gas molecules in a chamber. The likelihood that a reactive molecule will strike a particular target zone, such as a sensor surface that covers 5 cm² of a 50 cm² wall, if it is released at random, is 5/50 = 0.1, or 10%.

Signal strength and mobile coverage

Based on a mobile device's position and distance from signal towers, telecom companies use geometric probability to model the likelihood that the device will receive a strong signal.

For instance, if a circular area has a 2 km radius strong-signal zone inside a larger 5 km radius service area, the likelihood that a phone placed at random will be in the strong-signal area is

                                                           𝑃 =  (2)2 (5)2 = 425 = 0.16 

Manufacturing Quality Control 

By examining random samples within a batch, factories use geometric probability to find flaws. It is possible to model geometrically the probability of a product having a flaw in a particular area. For instance, geometric probability can estimate a 10% chance of a defective label if a machine cuts circular labels from a sheet where 10% of the area is prone to damages because of misalignment.