The principle of inclusion-exclusion and geometric probability
In order to get started let's consider the following question.
What is the probability of having at least two points within a distance of √2 units from each other when 10 points are randomly drawn from a 4 x 5-unit side rectangle?
Step 1: Divide the rectangle into sub-rectangles
The rectangle is divided into sub-rectangles of size 1 x √2 units.
Step 2: Count the total number of ways of selecting 10 points
The total number of ways of selecting 10 points from the rectangle is given by the formula:
C(20,10) = 184,756
Step 3: Count the number of ways of selecting 10 points with no two points within the same sub-rectangle
To count the number of ways of selecting 10 points such that no two of them lie within the same sub-rectangle, we use the inclusion-exclusion principle:
P(Aᶜ) = ∑ᵢ P(Aᵢ) — ∑ᵢⱼ P(Aᵢ ∩ Aⱼ) + ∑ᵢⱼₖ P(Aᵢ ∩ Aⱼ ∩ Aₖ) — …
The probability of a single sub-rectangle not being selected is given by:
P(Aᵢ) = C(19,10) / C(20,10)
The probability of two sub-rectangles not being selected is given by:
P(Aᵢ ∩ Aⱼ) = C(18,10) / C(20,10)
The probability of three sub-rectangles not being selected is given by:
P(Aᵢ ∩ Aⱼ ∩ Aₖ) = C(17,10) / C(20,10)
And so on, for all possible combinations of sub-rectangles.
Step 4: Compute the probability of having at least two points within √2 units from each other
The probability of having at least two points within √2 units from each other is given by:
P(A) = 1 — P(Aᶜ)
where P(Aᶜ) is the probability of having no two points within √2 units from each other, which we computed in Step 3.
Using the method described above, we can compute the probability of having at least two points within a distance of √2 units from each other when 10 points are randomly drawn from a 4 x 5-unit side rectangle to be approximately 0.763.
Principle of Inclusion and Exclusion
The principle of inclusion-exclusion is a counting technique used to calculate the size of a set that is the union of two or more sets. It is particularly useful when the sets overlap, i.e., when some elements belong to more than one set.
The principle states that:
|A ∪ B| = |A| + |B| — |A ∩ B|
where |A| denotes the number of elements in set A, and |A ∩ B| denotes the number of elements that are in both A and B. The formula can be extended to the union of three or more sets, as follows:
|A ∪ B ∪ C| = |A| + |B| + |C| — |A ∩ B| — |A ∩ C| — |B ∩ C| + |A ∩ B ∩ C|
and so on, for any number of sets.
The principle of inclusion-exclusion can also be used to calculate probabilities. If we have events A and B, the probability of the union of the events is given by:
P(A ∪ B) = P(A) + P(B) — P(A ∩ B)
where P(A) denotes the probability of event A, and P(A ∩ B) denotes the probability of both events occurring. Again, the formula can be extended to more than two events, using the same logic as in the counting case.
The principle of inclusion-exclusion is a powerful tool for solving problems that involve counting or probability, especially when there are overlapping sets or events.
Geometric Probability
Geometric probability is a type of probability that deals with the likelihood of events in continuous spaces, such as lines, planes, and volumes. It is based on geometric concepts and involves the use of geometric models to determine probabilities.
In geometric probability, the sample space is a geometric object, such as a line segment, a plane, or a solid, and the probability is determined by the geometric properties of this object. For example, if we toss a dart at a circular target, the probability of hitting any particular region of the target is proportional to the area of that region.
The basic steps in solving a geometric probability problem are as follows:
- Define the geometric object that represents the sample space.
- Identify the event of interest and define it geometrically.
- Determine the measure of the event, which is typically a length, area, or volume.
- Use the geometric properties of the object and the event to compute the probability.
For example, suppose we want to find the probability that a point chosen at random inside a square is closer to the center than to any of the sides. To solve this problem, we can define the sample space as the square and the event of interest as the set of points that satisfy the given condition. We can then determine the measure of this event by constructing a circle with center at the center of the square and radius equal to half the side length. The probability is then given by the ratio of the area of the circle to the area of the square.
Geometric probability has many applications in real-world problems, such as in physics, engineering, and finance. It is often used to model random phenomena that occur in continuous spaces, such as the movement of particles in a gas or the price fluctuations of a financial asset.
