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 \(n\),
$$
\lim_{n\rightarrow\infty} \big(1 - \frac{1}{n}\big)^{n} = \frac{1}{e}
$$
That fundamental number \(e\) shows up again!
Some good books to own to learn the art of probability
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Introduction to Probability Theory
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.
Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
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 \(n\),
$$
\lim_{n\rightarrow\infty} \big(1 - \frac{1}{n}\big)^{n} = \frac{1}{e}
$$
That fundamental number \(e\) shows up again!
Some good books to own to learn the art of probability
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms
This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Introduction to Probability Theory
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
Introduction to Probability, 2nd Edition
The Mathematics of Poker
Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.
Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
I might be missing something. Your answer seems to assume that the probability of a person getting out at any floor is independent of the floor, although this is not explicitly established in the setup. As that probability it 1/n it also seems to assume that the passengers know in advance how many floors there are, and are definitely planning to get out of the lift at some point (i.e. before it goes through the roof, or before they find themselves forced to get out at the top floor with probability 1), which seems to be at odds with the suggestion that they decide whether to get out only at the point that the lift has arrived at a floor.
ReplyDelete-Justin
Much clearer now - thanks! :-)
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