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A Shuffling Puzzle

Q: You have an array of items that you want to shuffle. You pick an element and then pick a random number uniformly between \([1,n]\) and swap it with the element following which you move on to the next element and repeat. Would this result in a fair shuffle?

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)

A: Some people shuffle a deck of playing cards in the above manner. They tend to pick a small (random sized) slab of cards from under the deck, place it on top and do so several times over.

The above shuffle would not result in a fair shuffle. Here is why. Assume you have \(n\) items in the list. When you pick an element and swap it out with another element there are \(n\) positions you could swap it with, including itself in which case there is no swap. This implies that the first swap would result in \(n\) arrangements. The second swap follows the same algorithm and would result in the same number of arrangements, that is \(n\). This leads to a total of \(n^{n}\) possible arrangements. However in a fair shuffle, the first element would be chosen from \([1,n]\), the second from \([1,n-1]\) and so on, leading to \(n!\) arrangements in total. Clearly, any strategy that gives a fair shuffle has to yield a number of arrangements \(N\) such that
$$ N \pmod{n!} = 0 $$
However
$$ n^{n}\pmod{n!} \ne 0 $$
As \(n^{n}\) does not divide \(n!\) it implies that the algorithm would end up favouring some combinations over the other resulting in a biased shuffle.

If you looking to learn probability & algorithms here are some good books.

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

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.

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

Quantum Poker
Well written and easy to read mathematics. For the Poker beginner.

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

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