midProbability
What is the intuition behind law of total expectation?
Updated May 17, 2026
Short answer
It states that overall expectation is the weighted sum of conditional expectations.
Deep explanation
E[X] = E[E[X|Y]]. This means we can compute expectation by first conditioning on another variable and then averaging. It simplifies complex probabilistic computations by breaking them into conditional parts.
Real-world example
Expected salary computed across departments.
Common mistakes
- Ignoring conditioning structure and computing directly.
Follow-up questions
- Why does it work?
- Where is it used?