Strings Puzzle

Q. A jar has N strings in it. You randomly pick up two loose ends and tie them up. You continue doing this until there are no more free ends. What is the expected number of loops in the jar?

A. This is another one of those puzzles which gets vastly simplified when you place it in a recursive perspective. A diagram with 3 strings would best explain the scenario. Ref figure below.

If there are 'n' strings, there are 2n loose ends to pick up from. The first loose end can be picked up randomly. While picking the second loose end, there are two possibilities.

1. It is part of the same string you just picked up OR
2. It is the loose end of another string.

For 1), the probability of this event happening is

while for 2) it is 1 minus the above, giving

This implies that with probability 1/(2n-1), the expected number of strings will be 1 + P(n-1) and with probability (2n-2)/(2n-1) it will be P(n-1). This leads to the recursive formulation of the expected number of loops to be

If you re-arrange the terms, the P terms cancel out nicely yielding

This result converges to log(2n-1)

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Here are a few
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

Let There Be Range!: Crushing SSNL/MSNL No-Limit Hold'em Games
Easily the most expensive book out there. So if the item above piques your interest and you want to go pro, go for it.

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

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.