Law of Total Expectation Interview Questions
How to use the law of total expectation in quant interviews by splitting hidden states, cases, and first-step outcomes.
Candidates practicing expected value and conditional probability together.
The idea in words
The law of total expectation says an overall average can be built from conditional averages. Split the world into cases, find the expected value in each case, then weight by how likely each case is.
Use a complete partition
The cases should cover every possibility without overlap. Hidden states, first draws, first rolls, and signal outcomes are common interview partitions.
Concrete example
Suppose a game first chooses a fair coin. If heads, the payoff has expected value 10; if tails, it has expected value 4. The total expected payoff is 0.5 x 10 + 0.5 x 4 = 7.
Connection to conditional expectation
Conditional expectation solves each case. Total expectation combines those case-level answers into one overall expected value.
When to use it
Use total expectation when the first event changes the rest of the problem. It is especially useful for two-stage games, hidden coins, conditional draws, and recursive expected values.
Common mistakes
Candidates often average case expectations without weighting them. The weights are the probabilities of the cases, not the number of cases written down.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.