Q: A large number of people \(n\) get into an elevator in a skyscraper building which has the same number of floors as \(n\). They have predetermined at what floor they will step out completely at random. Once they step out, the elevator proceeds upwards. What is the probability that nobody steps out at the 3rd floor.

Probability Theory: The Logic of Science

A: In a later post, I'll describe an abstraction that can be used to solve such problems and similar ones. The key is to be able to map statistical problems one runs into to this abstraction (placing \(m\) balls in \(n\) bins).

The probability that a person gets off at floor 3 is \(\frac{1}{n}\) where \(n\) is the number of floors in the building. The probability that the person does not get off at the 3rd floor is thus \(\big(1 - \frac{1}{n}\big)\). The probability that none of the \(n\) people step out at the 3rd floor is given by

$$

\big(1 - \frac{1}{n}\big)^{n}

$$

In the limiting case, for large \(n\),

$$

\lim_{n\rightarro…

Probability Theory: The Logic of Science

A: In a later post, I'll describe an abstraction that can be used to solve such problems and similar ones. The key is to be able to map statistical problems one runs into to this abstraction (placing \(m\) balls in \(n\) bins).

The probability that a person gets off at floor 3 is \(\frac{1}{n}\) where \(n\) is the number of floors in the building. The probability that the person does not get off at the 3rd floor is thus \(\big(1 - \frac{1}{n}\big)\). The probability that none of the \(n\) people step out at the 3rd floor is given by

$$

\big(1 - \frac{1}{n}\big)^{n}

$$

In the limiting case, for large \(n\),

$$

\lim_{n\rightarro…