First-Step Analysis in Probability Interviews
First-step analysis interview prep for conditioning on the next move, setting state equations, solving probabilities, and expected values.
Candidates working on random walks, Markov chains, dice games, and stopping-time prompts.
Condition on the next move
First-step analysis solves a process by conditioning on what happens immediately next. The next move sends the problem into a new state.
Use states, not stories
Translate the prompt into states such as current position, current score, current streak, or current boundary distance. Then write equations between states.
Probability equations
For hitting probabilities, the probability from a state is a weighted average of hitting probabilities from the possible next states.
Expectation equations
For expected time, add the first step you take now, then add the weighted expected remaining time from next states.
Concrete setup
If a random walk moves left or right with equal probability, the value from position i can be written using the values from i - 1 and i + 1.
Common mistakes
Candidates often forget the plus one in expected-time equations. Probability equations and time equations have different structures.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.