Recently I came across a book “How not to be mistaken. The Power of Mathematical Thinking ”, by Jordan Ellenberg - this is what will be discussed.
In this article, I want to provide some simple math / logic problems that:
- Will help to stretch your brains a little;
- Perhaps someone will be helped to learn new concepts from mathematics and statistics.
Well, and to advise an interesting book, which, with examples from real life, shows what mistakes can be made even in simple situations, and how to ask yourself the right questions so as not to make these mistakes in the future.
Are you interested? Then they drove under the cut!
Small introduction
All the problems presented in this article are nothing more than selected problems posed to the reader by mathematician Jordan Ellenberg, author of the book How not to be mistaken. The power of mathematical thinking. "
In his book, the author on 512 pages in simple language without formulas and complex terms tries to show the reader that mathematics is not "learn the multiplication table" and "formulas by which you can take the integral", but a certain view of the world, which can be very useful in different situations.
I want to immediately point out some of the features of this book that I was able to notice.
First nice feature
Quite often, creations that claim to be "something difficult is explainable to everyone" have a common problem - they begin to chew on very elementary things on the simplest examples, completely omitting terms or "too complex concepts." Ellenberg was able to find a certain balance. That is, if you need to introduce an axiom from geometry, he will introduce it and write it down. If we are talking about the law of large numbers, then he will lead the formulation of the law. It will simply lead to her smoothly and through a life example.
The second "catchy" feature
There are a lot of references to real articles / research / events. Here you can find examples from the Second World War, and the assessment of higher educational institutions in America, and the essence of some gambling, and fraudulent schemes.
The third feature that I would call specific
There are many discussions on various topics: from religion to the correct structure of the state. Personally, I don't really like such moments. Here's a quote from the book as an example to make it easier to explain what I'm talking about:
As much as I love numbers, I believe that people should adhere to some of the principles: "I don't believe in God," "I believe in God," or simply, "I'm not sure." As much as I love Bayesian inference, I believe that people are better off finding faith (or dropping it) without resorting to numbers. In this matter, mathematics is silent.
But if you evaluate the book as a whole, it is quite interesting and exciting.
Who will benefit from reading the book?
It seems to me that, first of all, the book will be useful to those who do not deal with statistics directly, but look at the data every day and try to ask themselves (or someone) the right questions to make decisions. At least because there is a whole block about ways to juggle numbers and how to notice and recognize this juggling of overt data manipulation.
Also, the book can be useful for Junam analysts. After all, it is always better to learn from someone else's mistakes than to make your own. And the author of the book cites a lot of such “perfect mistakes”.
And, of course, for those who just like to study statistics on real problems. In this article, I tried to convey the format of the narration that the author adheres to, namely the format: task → incorrect reasoning → a little theory → correct answer.
You can also safely slip this book to your parents or friends who are far from the world of mathematics and constantly torment you with questions: "What are you doing?" or “oh, why go to a technical college? The calculator and the computer will calculate everything for us! "
Tested personally on my own parents, by the way.
Well, when are the tasks already?
We're almost done with an introduction and a brief description of the book itself, so let's move on to the fun part.
Problem number 0 (warm-up): "What to strengthen on the plane?"
Imagine that the irreparable happened and you find yourself in 1943 in the midst of World War II. You are an engineer whose task is to improve existing aircraft so that they are more resistant to enemy attacks.
But there is a limitation: additional armor is heavy, which means that covering the entire aircraft is not an option, it will become heavy and less maneuverable.
Your task: to find the optimal solution.
You have at your disposal data on the structural elements of the aircraft and the number of bullet holes per square foot, which are collected from the examination of aircraft returning from combat battles.
- Engine - 0.11
- Fuselage - 1.73
- Fuel system - 1.55
- The rest of the aircraft - 1.8
Well, common sense.
Attention, the question: what should be strengthened first?
Answer
Problem # 1: "Should I trust a stock broker from Baltimore?"
Imagine checking your mail on Monday and stumbling across a letter from a stock broker in Baltimore that says that stock in Company X will go up this Monday.
And, lo and behold, it happens.
Next Monday and again a letter from this broker, but this time the letter says that Y shares will start to fall.
And again the prediction comes true.
History repeats itself ten Mondays in a row. On the eleventh Monday you receive a letter where you are offered to entrust your money to the fund of this broker. Yes, the commission is an order of magnitude higher than everyone else's, but the “unique scheme and foresight” is worth it.
Attention, question: should you trust a stock broker from Baltimore? Where could there be a catch?
Answer
Problem number 2: "Will everyone suffer from excess weight?"
The National Health and Nutrition Examination Program monitors health data from a large, representative sample of citizens. In particular, data on the proportion of people who are overweight.
After examining the data, the authors of the article in Obesity magazine made the assumption: By 2048, the entire population of America will be overweight.
Attention question: is it so? If not, where is the error in reasoning.
Answer
Problem number 3: "Who is the best to flip coins?"
. . , . .
, , . 10 100 , — 100 100 .
?
...
In general, the first part of the book consists almost entirely of tasks of this kind. The author introduces basic mathematical concepts and even gives the logic of proving some theorems.
Smoothly approaching the second part, the complexity of the concepts increases along with the complexity of the tasks that the author suggests. But everything is also simple and understandable even for a person who has not attended 100,500 hours of lectures on higher mathematics.
You gradually move from survivor bias to regression, to mean. After reading a couple more chapters, you already know about Berkson's mistake. You master a few more sections, and now you are getting close to Hilbert's problems. This is how you travel through the world of mathematics and statistics and learn about fundamental concepts and theorems. For example, with the null hypothesis, with Gödel's incompleteness theorem, with Buffon's problem of throwing a needle, with the theorem on the distribution of prime numbers.
So if you suddenly feel that your device responsible for critical thinking is a little junk, or you start to forget the program of mathematical statistics, then the book will do as evening reading for a wine.
What else to read?
If you are generally interested in such books, in which complex things from statistics, analytics and the world around you are explained in simple language, then pay attention to this selection:
1. Statistics and cats. Author: Vladimir Saveliev.
Unexpectedly, but a book about statistics on the example of cats. On cats, well-known concepts are introduced, such as fashion, average, median. On seals, they talk about variance and standard deviation. Stat.criteria are analyzed on cats. In general, everything is on cats and everything is super-simple (and even with pictures).
Example illustration from the book
Recently, our team held an A / B testing workshop at the company. During the preparation process, they often turned to this book to spy on an easy way to explain something.
2. Freakonomics: The bully economist and daredevil journalist explore the hidden causes of everything. Authors: Stephen Levitt, Stephen Dubner.
Which is more dangerous - a firearm or a swimming pool? Why do drug dealers continue to live with their parents? How much does parenting and choosing a name really mean to a child?
These are not strange questions, these are chapter titles from the book. It is these questions that are sorted out in the book, and all this is supported by references to serious scientific research.
In general, the book teaches you to be critical of any incoming information, think logically and notice small details.
And if you want more?
If you want to take a more systematic approach to the study of statistics or get deeper into analytics, then catch several courses and collections of lectures.
1. A course on the basics of statistics on stepik . The course is free. Suitable even for beginners, as lectures start from the very basics.
2. GoPractice - a course about product management based on analytics and data. Paid.
The entire learning process is based on the fact that you "live" the professional history.
The learning process follows the scheme:
- There is a problem that needs to be solved (and which may well be encountered in real life);
- Several tasks in this topic;
- Theory;
- Several tasks for testing this theory.
3. Unexpectedly, but institute lectures - Statistics, applied stream, MIPT . Hardovo, not for beginners, but thoroughly.
Link to YouTube .
What books / courses would you recommend reading?