Probability
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February 14, 2026
Elevator Probability
You enter a building with 12 floors and press your floor at random. What is the expected number of stops if 3 other passengers do the same?
You step into an elevator in a 12-story building. You and 3 other passengers each press a floor button independently and uniformly at random (floors 1–12, excluding the ground floor you entered on — so 11 choices each).
The Challenge
What is the expected number of distinct floors the elevator will stop at?
Bonus: How does your answer change if passengers can also choose to exit on the ground floor?
💡 Hint
Use the complement: E[stops] = 11 - E[floors not chosen]. Use linearity of expectation. Solution
By linearity of expectation: E[stops] = 11 × P(a given floor is chosen by at least one person) = 11 × (1 - (10/11)⁴) ≈ 11 × 0.336 ≈ 3.70 stops.