How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg

Summary

'How Not to Be Wrong: The Power of Mathematical Thinking' by Jordan Ellenberg is an engaging exploration of how mathematical thinking influences our everyday lives, often in surprising ways. Ellenberg uses humor and real-world examples to demonstrate how math helps us make better decisions, avoid mistakes, and see the world more clearly. The book covers concepts ranging from statistics and probability to geometry, showing how these abstract ideas play a practical role in domains like politics, medicine, and finance.

Life-Changing Lessons

1. Mathematical thinking can uncover hidden truths and prevent us from making logical errors in daily life.

2. Understanding statistics protects us from being misled by flawed data, misleading advertisements, and poor policy decisions.

3. Probability helps us better evaluate risk, improve our decision-making, and recognize the limits of certainty.

Publishing year and rating

The book was published in: 2014

AI Rating (from 0 to 100): 93

Practical Examples

1. Abraham Wald's Bullet Holes

   Ellenberg discusses how Abraham Wald analyzed where returning WWII planes were hit by bullets. Instead of reinforcing the panels that showed the most damage, Wald realized that the unharmed spots were exactly where reinforcements were needed, as planes hit there didn’t return. This example illustrates the concept of survivorship bias.

2. The Obesity Paradox

   The book explores how statistical measures can be misleading, as in the case of the 'obesity paradox', where overweight patients seemed to survive certain illnesses at higher rates. Ellenberg breaks down the statistical flaws and possible confounding variables to demonstrate why correlation does not always imply causation.

3. The Laffer Curve

   Ellenberg dives into the Laffer Curve, an economic theory suggesting there is an optimal tax rate that maximizes revenue. He critiques the misuse of simplistic mathematical models in government policy by showing how assumptions can lead to flawed or overly confident conclusions.

4. Deadly DNA Tests

   The book reviews how misunderstanding probability leads to errors in interpreting DNA evidence in criminal trials. Ellenberg explains the prosecutor's fallacy, where the chance of a DNA match is confused with the actual likelihood of guilt, potentially resulting in wrongful convictions.

5. Lottery Strategies

   Ellenberg uses lottery mathematics to show how understanding the odds can demystify common misconceptions about luck and probability. He examines cases where savvy groups gamed state lotteries by exploiting ticket structures, emphasizing the value of mathematical scrutiny.

6. Regression to the Mean

   Through stories of sports statistics, Ellenberg explains regression to the mean, a phenomenon where extreme outcomes are likely to be followed by more typical ones. Failing to recognize this can lead to false beliefs in phenomena like 'the hot hand' in basketball.

7. Math in Voting Systems

   Ellenberg examines how different voting systems can produce wildly different outcomes, even with the same voter preferences. He uses mathematical tools to analyze fairness and the potential for paradoxes in election results.

8. Helicopter Parenting and Statistical Outliers

   Ellenberg questions whether extreme parental involvement actually leads to academic success, showing how statistical outliers can tempt us to draw unwarranted conclusions about cause and effect.

9. Medical Testing and False Positives

   The book delves into why even highly accurate medical tests can yield substantial numbers of false positives when testing for rare diseases. Ellenberg uses Bayes' Theorem to help readers understand how to interpret such test results.