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Q: You play a game with a friend where he chooses two random numbers between 0 and 1. Next you choose a random number between 0 and 1. If your number falls between the prior two numbers you win. What is the probability that you would win?
A: Consider the number line between 0 and 1 shown in figure below
Let \(x\) and \(y\) be the two numbers chosen. The probability of a win, i.e. choosing a number in the range between the two chosen numbers can be estimated as \(\frac{|y-x|}{1}\). Note, we have chosen the modulus operation because \(x\) could be greater than \(y\) or vice versa. The feasible region of numbers to be chosen to win, is the ratio of the absolute difference between \(x\) and \(y\) divided by the total possible range, which is 1. In order to estimate the probability that a third chosen number will lie between the two we integrate out (a double integral) between the ranges of \([0,1]\). This is estimated as
$$
P(\text{win}) = \int_{0}^{1}\int_{0}^{1}|y - x| dx dy
$$
To evaluate the above integral, we split the inner integral as follows
$$
P(\text{win}) = \int_{0}^{1}\Big[\int_{0}^{y}(y - x)dx + \int_{y}^{1}(x-y)dx\Big]dy
$$
The inner integral evaluates to \(y^2 - y +\frac{1}{2}\) which on further integration between the ranges of \([0,1]\) yields
$$
P(\text{win}) = [\frac{y^3}{3} - \frac{y^2}{2} + \frac{y}{2}]_{0}^{1} = \frac{1}{3}
$$
which is the sought probability.
If you are looking to buy some books in probability here are some of the best books 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
Overall an excellent book to learn probability, well recommended for undergrads and graduate students
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.
Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.
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 and even 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
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"
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.
A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online
Probability and Statistics (4th Edition)
This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good
An alternative solution: the problem can be rephrased as "Draw three points in [0,1] at random. What is the probability of the third point falling between the first two?". Ordering the chosen points by magnitude results in one of 3!=6 possible permutations. By symmetry, each permutation is equally likely to occur. The event we are interested in is comprised of two of the permutations ("1-3-2" and "2-3-1"). Therefore the probability of this event is 2/6 = 1/3.
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