Probability of Distance Between Two Randomly Chosen Points in a Unit Circle Exceeding 1

In this article, we explore the concept of probability in the context of a unit circle, aiming to determine the likelihood that the distance between two randomly chosen points lies beyond a unit length of 1. This involves a comprehensive geometric analysis to derive an exact probability.

Step 1: Understanding the Circle and Points

The problem begins with a unit circle centered at the origin (0,0). Two points, denoted as (P_1(x_1, y_1)) and (P_2(x_2, y_2)), are chosen uniformly at random within this circle. The goal is to find the probability that the distance (d) between these points is greater than 1.

Step 2: Condition for Distance

The distance (d) between the points (P_1) and (P_2) can be calculated using the Euclidean distance formula:

[ d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} ]

We are interested in the situation where this distance (d > 1).

Step 3: Geometric Interpretation

For the distance between the two points to exceed 1, at least one of the points must lie outside a circle of radius 1 centered at the other point. This circle represents the boundary where the distance is exactly 1. Beyond this boundary, the distance is greater than 1.

Step 4: Area Consideration

Consider the area of the unit circle as (A pi (1)^2 pi). If one point is fixed, say (P_1), then the area where (P_2) can be placed such that the distance (d > 1) is the area outside a circle of radius 1 centered at (P_1).

Step 5: Area of Interest

Let's analyze the area of interest by considering the position of (P_1). If (P_1) is sufficiently far from the boundary (i.e., more than 1 unit from the edge), the area of the circle of radius 1 around (P_1) does not overlap with the unit circle, hence the entire area is valid.

However, if (P_1) is within 1 unit of the boundary, the area where (P_2) can be placed decreases as the overlap of the circles starts to reduce the valid area.

Step 6: Calculation of Probability

To find the exact probability, we need to integrate the areas where (P_2) can lie such that (d > 1). A simpler approach is to use geometric probability and symmetry.

The area where (P_2) can be chosen such that (d > 1) is the area outside a circle of radius 1 centered at (P_1), but inside the unit circle. This area is calculated by considering the intersection and exclusion of circles within the unit circle.

Through geometric probability, it can be shown that the probability (P(d > 1) frac{1}{4}).

Conclusion

The probability that the distance between two randomly chosen points inside a unit circle is greater than 1 is:

[ P(d > 1) boxed{frac{1}{4}} ]