Drunk on the Number Line

What this random-walk return probability question tests

This is an easy probability question that appears frequently in quant interviews because it combines intuitive setup with a surprising result. It asks you to reason about the long-run behaviour of a symmetric random walk and challenges the assumption that symmetry guarantees return.

To approach this, you need to think carefully about what "ever return" means over infinitely many steps, set up a recursive probability equation, and solve it. The question rewards clear reasoning about infinite sequences and conditional probability, rather than heavy computation. Many candidates find the intuition counterintuitive, which is precisely why firms use it to separate solid probabilistic thinking from superficial pattern-matching.

• Infinite-step probability and taking limits
• Recursive probability equations and self-similarity
• Symmetry arguments and their limits