Probability Calculation for Intersecting Circles within a Unit Square

When randomly taking two points on the boundary of a unit square and drawing two circles with these points as centers and radii of 1/2, what is the probability that these circles intersect? This problem requires a careful analysis of the geometric constraints and the application of probability principles.

Case Analysis for Circle Intersection

Vertices of the Square: When the two points are on the same side of the square (probability 1/4), the circles will intersect with a probability of 1. Opposite Sides: When the two points are on opposite sides of the square (probability 1/4), the circles will intersect with a probability of 0. Adjacent Sides: When the points are on adjacent sides (probability 1/2), the intersection probability involves a more detailed calculation.

Adjacent Sides Case

Let's denote the points as A and B, and the common vertex of the two adjacent sides as V. The distances of points A and B from vertex V are x and y, respectively.

For the circles to intersect, the distance between their centers (A and B) must be less than or equal to 1 (the sum of their radii). This distance is given by the Euclidean distance formula:

[ d(A, B) sqrt{x^2 y^2} leq 1 ]

The condition for the circles to intersect thus becomes:

[ x^2 y^2 leq 1 ]

Given that the points A and B are on adjacent sides, the distance y can be expressed as a function of x: ( y sqrt{1 - x^2} ).

Therefore, the probability that the circles intersect can be calculated by integrating over the possible values of x and y from 0 to 1:

[ P(text{Intersection}) int_{0}^{1} sqrt{1 - x^2} , dx ]

This integral can be solved using standard techniques. The result of the integral is (frac{pi}{4}), which represents the probability that the circles intersect when the points are on adjacent sides.

Overall Probability Calculation

Combining the probabilities from all the cases, we have:

[ P frac{1}{4} cdot 1 frac{1}{4} cdot 0 frac{1}{2} cdot frac{pi}{4} ]

Substituting the values:

[ P frac{1}{4} 0 frac{pi}{8} frac{pi}{8} frac{1}{4} ]

Approximating (pi approx 3.14159), we get:

[ P approx frac{3.14159}{8} frac{1}{4} 0.392699 0.25 0.642699 approx 0.643 ]

Thus, the overall probability that the two circles intersect is approximately 0.643.

Conclusion

The detailed calculation confirms the approach and ensures that the probabilities from each case are correctly combined to give the final probability of the circles intersecting.

Key Points: The problem involves a combination of geometric probability and integration. The integration of (sqrt{1 - x^2}) over the interval [0, 1] is a crucial step in determining the probability for the case where the points are on adjacent sides.

Keywords: probability of intersecting circles, unit square, intersection probability calculation.