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The numbers x1, x2, x3, and x4 are chosen at random in the interval [0,1]. Let I be the interval between x1 and x2, and let J be the interval between x3 and x4. Find the probability that intervals I and J overlap.

Aug 6, 2024

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Probability of Overlapping Intervals

Understanding the Problem

We have four random points (x1, x2, x3, x4) on the interval [0, 1]. We form two intervals: I from x1 and x2, and J from x3 and x4. We want to find the probability that these intervals overlap.

Approach

Instead of directly calculating the probability of overlap, it's often easier to calculate the probability of the complement: the probability that the intervals do not overlap.

Non-Overlapping Intervals

For the intervals I and J to not overlap, one must be completely to the left of the other.

There are two possibilities:

x2 < x3: Interval I is completely to the left of J.

x4 < x1: Interval J is completely to the left of I.

Geometric Interpretation

We can visualize this problem in a 4-dimensional space where each axis represents one of the x values. However, due to the symmetry of the problem, we can reduce it to a 2-dimensional space by considering the pairs (x1, x2) and (x3, x4).

Each pair (x1, x2) and (x3, x4) can be represented as a point in the unit square [0, 1] x [0, 1]. The condition x2 < x3 corresponds to the triangle below the line y = x, and the condition x4 < x1 corresponds to the triangle above the line y = x.

The total area of these two triangles is 1/2.

Final Calculation

Since the total probability space is the unit square with area 1, the probability of non-overlapping intervals is 1/2.

Therefore, the probability of overlapping intervals is 1 - 1/2 = 1/2.

So, the probability that intervals I and J overlap is 1/2.

Aug 8, 2024