Two stains, many pieces

What this uniform-random partitioning probability question tests

This is a medium-difficulty probability question that combines geometric reasoning with combinatorial counting. It asks you to find the probability that two fixed random points remain in the same segment after a bar is randomly partitioned into many pieces.

To solve problems of this type, you need to think carefully about how the initial marked points and the subsequent break points interact. The key is setting up the problem correctly: understanding what configurations lead to the marked points being grouped together, and then integrating or counting over the joint distribution of all random positions. Many candidates benefit from drawing a diagram and considering whether order matters.

• Uniform distributions on continuous intervals
• Conditional probability and independence
• Partitioning and segment membership
• Symmetry and exchangeability