Bumping into Cars

What this collision-clustering probability question tests

This is a medium-difficulty probability question that asks you to reason about the equilibrium structure of a dynamic system. It appears frequently in quant interviews because it rewards both creative problem setup and careful analytical thinking—and because the closed-form answer is elegant once you see the right angle.

The key insight is recognizing what determines whether two adjacent railcars will merge into the same clump. Rather than simulating collisions, you need to identify the stable configuration and count its expected size. This typically involves conditional reasoning: given the relative ordering of speeds, which pairs will eventually collide?

• Linearity of expectation and indicator random variables
• Conditional probability and exchangeability
• Recognizing Markov structure in sequential processes

Once you derive the analytical form, the problem permits use of a calculator to evaluate the final numeric answer to three decimal places.