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Cracking Numbers in Envelopes

Q: You are shown two envelopes with numbers written inside them. The numbers are chosen uniformly randomly from the range of 1 to a 100. You are allowed to open one envelope and look at the number inside following which you need to guess if the number in the other envelope is bigger or smaller of the two. You win if your guess is correct. What should your strategy be to have a greater than 50% win probability?

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everone Else!)

A: This puzzle is similar to the two envelopes paradox. At first pass, it appears that it is impossible to do better than 50% in win probability. But surprisingly, it is possible to do better. What is even more surprising is that it is an amazingly simple solution! All you need to do is to pick a function that is monotonically increasing in the domain 1 to 100. For example, let \(X\) be the number in the envelope you have opened. Let us make a function which takes in the number \(X\) as input and the can be eventually mapped to a range of \([0,1]\) so it "looks" like a probability estimate: A simple one would be to divide \(X\) by a 100.
$$
f(x) = \frac{x}{100}
$$
Next, simply declare the chosen number \(X\) to be the bigger of the two if \(f(X) > 0.5\). This strategy always yields a win rate greater than \(\frac{1}{2}\). Do you see why this works? Lets say the chosen number is 66, you would conclude that you have a \(66\%\) chance that the chosen number is the greater one and your guess would be right \(66\%\) of the time. Likewise, lets choose a function that is not so trivial looking, but continues to be monotonically increasing, for example
$$
f(x) = \frac{x^2}{\pi}
$$
Now map the value returned to a probability as
$$
P(X) = \frac{f(X)}{f(100) - f(1)}
$$
and declare the chosen number to be the greater if \(P(X) > 0.5\). The reason this works is because for a monotonically increasing function \(f(X_1) > f(X_2) \text{ if } X_1 > X_2\), it is as simple as that.

If you are looking to learn probability, some good books to own are shown below

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.


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