Leonard Mlodinow on probability blindspots

In 2002, psychologist Daniel Kahneman won a Nobel Prize in Economics for work that pointed out that human beings often have problems reasoning through real world problems involving probabilities.

Caltech physicist Leonard Mlodinow has written a fascinating and clearly written new book, The Drunkard's Walk: How Randomness Rules Our Lives, with many beautiful examples illustrating common logical fallacies. You can learn a good deal in a very enjoyable way by reading his book. You might want to start out by watching a talk he gave about his book to Google employees. (See the video above.)

As Mlodinow points out, probability blind spots can cause seriously bad decisions in many domains. Many professionals, including physicians, judges, and investors, make errors in reasoning through situations involving probabilities. (Sadly, it appears that medical schools and law schools don't teach much about probability. Business schools DO teach about probabilities, but it's not clear how much actually sinks in.) I personally think the answer is that students need to grow up thinking hard and deeply about probability. The habits of thinking correctly about probabilistic calculations need to be ingrained deeply in all of us long before we become jurors or adult patients, let alone judges or physicians.

There is a growing consensus about the need for a really sound mathematical education in probability and statistics. Harvey Mudd math professor Art Benjamin makes a good case for it in his TED Talk video below:


Hat tip: Richard Rusczyk

Physics blind spots

Via Uncertain Principles, here's some fascinating evidence about a physics blindspot.

Number sense!

How Many Millions are in a Trillion? from Econ4U on Vimeo.

The 21% correct response figure cited at the end of the video comes from a multiple choice survey given to a thousand American adults. A monkey throwing darts at the five answer choices they provided in the multiple choice version of their poll would hit the correct answer 20% of the time, so a 21% correct response rate is discouraging.

Search for Intelligent Life points out another video describing the public's bewilderment in the face of large numbers:



Edith Stokey and Richard Zeckhauser have a simple and sensible suggestion for getting a handle on large numbers: long division.

A useful number to keep in mind for back-of-the-envelope calculations is that the US population is roughly 300 million.

So the next time you read that the federal government is considering spending a billion dollars on something, you can think that's over $3 per American. And when the federal government considers spending a trillion dollars on something, that's over $3K per person.

Silly standardized test question puzzler

Tanya Khovanova posed an interesting puzzle from VI Arnold:

I have a tiny book written by Vladimir Arnold Problems for Kids from 5 to 15. A free online version of this book is available in Russian. The book contains 79 problems, and problem Number 6 criticizes American math education. Here is the translation:

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle. American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

This one is worth thinking about.

The Russian students to whom Professor Arnold posed this problem were probably members of math circles, which originated in Eastern Europe and Russia and encourage outside-the-box thinking rather than cookbook approaches to problems.
Why couldn't the Russian students solve this apparently straightforward and simple problem?

Scroll down this page for the explanation.





















The Russian students realized that the triangle specified in the problem can't exist. The altitude to the hypotenuse of a right triangle can never be more than half the length of the hypotenuse. In fact, the altitude to the hypotenuse of a right triangle will always be exactly half the length of the hypotenuse. This fact is easy to see if you recall that every right triangle can be inscribed in a semicircle.

It's troubling to think that the American standardized testing company allowed this nonsensical problem to stay on their test for ten years!

MAA Minute Math and the AMC problem database

If you want to "Exercise your mind daily with a problem from the AMC-8, AMC-10, or AMC-12, provided by MAA's American Mathematics Competitions," visit the MAA Minute Math website.

Note that the official solutions provided are correct, but are not always the most efficient, so it's very worthwhile to think about more creative approaches of your own.

Minute math also provides interesting data on the percentage of test-takers who got each question correct. If you get the question wrong, you can take comfort in the fact that you generally have lots of company! (And bear in mind that the sample of AMC contest-takers is not a random sample of the population. The pool of students who take AMC exams includes a disproportionate number of strong and enthusiastic math students.) Whether you got the question right or wrong, it's interesting to look at the "distractor" wrong answers, which often give insights into common misconceptions.

It's important not only to understand your own blind spots, but also to recognize the blind spots of other people. This is true both in math competitions (which often have team-based components, so it's important to anticipate areas in which teammates may make mistakes), and also in the real world (perhaps the financial bubble which led to the recent meltdown would have been "pricked" sooner if more financial decision makers had considered the blind spots of other financial decision makers.)

If you want to look at the entire database of problems used on the AMC8/10/12 contests over the past decade, they are available here in a format that allows you to select particular types of problems, for example, discrete math, geometry, etc.