The Ballot Problem

What this ballot-problem probability question tests

This is a medium-difficulty probability question that appears in quantitative finance interviews to assess whether a candidate can reason about path-dependent probability and combinatorial counting. Rather than asking for a final outcome, it probes the likelihood of a specific sequence of states—a skill central to Monte Carlo simulation, Markov chains, and risk-path analysis.

The question requires you to count favorable orderings among all possible orderings, but with a constraint: one candidate must maintain a strict lead throughout the entire count, not just at the end. This demands clear thinking about when an ordering is valid and how to enumerate or derive valid paths. Most candidates benefit from drawing out small examples, identifying the combinatorial structure, or recognising a classical result.

• Counting orderings subject to inequality constraints
• Path-counting in lattice problems
• Distinguishing between final outcomes and intermediate states