The Broken Sequence

What this hard probability question tests

This is a hard probability question that combines Markov chains and expected value in a way common at quantitative trading and research interviews. It rewards candidates who can recognize a memoryless stopping problem, set up the right state space, and solve a system of linear equations without getting lost in casework.

The key insight is that your state at any point depends only on the most recent roll, not the full history. From each state, you must compute the probability of stopping immediately and the probability of transitioning to each successor state. Strong solutions explicitly define states (die values), write out the recurrence relations for expected rolls from each state, and solve the resulting system cleanly.

• Markov property and state-space design
• Stopping times and first-passage analysis
• Systems of linear equations in expected value
• Probability of ordered comparisons on discrete uniform outcomes