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

Summary

In this engaging talk, a math professor narrates intriguing stories to illustrate the power of mathematical thinking. From Abraham Wald's strategic WWII armor placement on planes to analyzing the Massachusetts Cash Windfall lottery, the professor demonstrates how math is not just about computation but also about asking the right questions and challenging assumptions. The talk dives into a detailed analysis of a lottery exploit by MIT students, showcasing how a deep understanding of probability and combinatorial designs can yield significant advantages, turning players into the 'house' in a game of chance.

Takeaways

• 📚 The speaker is a math professor who enjoys discussing math beyond the classroom setting and shares stories from the book 'How Not to Be Wrong'.
• 🔢 The story of Abraham Wald highlights the importance of mathematical thinking in problem-solving, especially in the context of World War II where Wald advised to armor the parts of planes not hit by bullets, as those that were hit did not return.
• 🎓 The purpose of mathematical thinking extends beyond computation to include critical questioning of assumptions and the ability to reframe problems.
• 🎲 The Massachusetts Cash Windfall lottery story demonstrates how a change in rules can drastically alter the dynamics of a game, leading to unintended consequences.
• 🏆 The roll down rule in the Massachusetts lottery, designed to boost player interest, inadvertently created a scenario where high-volume players could calculate a positive expected value and profit from the game.
• 🤓 MIT student James Harvey and others exploited the roll down rule by calculating the expected value of tickets and buying in bulk, effectively becoming the 'house' in their own right.
• 🏛️ The state of Massachusetts inadvertently benefited from the high-volume players, as the increased ticket sales generated significant state revenue.
• 💡 The comparison between the lottery strategy and a casino operation illustrates that the high-volume players were not cheating but rather using a mathematical advantage.
• 📉 The story of the lottery ended when the scheme was exposed by the media, leading to a cessation of the profitable roll down days as public perception changed.
• 🔍 The MIT group's choice to fill out lottery tickets by hand rather than using a quick pick machine suggests a strategic approach to minimize risk, possibly related to combinatorial designs.
• 📐 The connection between the lottery strategy and geometric concepts like the Fano plane and projective geometry hints at a deeper mathematical framework that could be used to optimize ticket selection.

Q & A

Who is the speaker in the transcript and what is their profession?

The speaker is a math professor who has written a book titled 'How not to be wrong' and is discussing mathematical concepts and stories related to it.

What is the significance of Abraham Wald's story in the transcript?

Abraham Wald's story is used to illustrate the importance of mathematical thinking, which involves not just computation but also questioning assumptions and sometimes overturning the original question, as Wald did with the plane armor problem.

What was the role of the Statistical Research Group (SRG) during the war mentioned in the transcript?

The SRG was a top-secret installation where top mathematicians and statisticians worked on problems related to the conduct of the war, particularly those of a mathematical nature.

What is the concept of 'expected value' in the context of the lottery discussed in the transcript?

The 'expected value' is a mathematical term used to describe the average amount won per lottery ticket over a large number of plays. It is calculated by considering the probability of winning and the amount won in relation to the cost of the ticket.

Why did the Massachusetts lottery introduce the 'roll down rule'?

The 'roll down rule' was introduced to prevent demoralization among players when no one won the jackpot for an extended period. It allowed the jackpot money to roll down into lower-tier prizes, making them more attractive and encouraging continued participation.

What was the unintended consequence of the 'roll down rule' in the Massachusetts lottery?

The unintended consequence was that it created a situation where the expected value of a lottery ticket on a roll-down day was higher than its cost, making it a profitable investment for those who recognized this and bought tickets in bulk.

Who were the main players that took advantage of the Massachusetts lottery's 'roll down rule'?

The main players were a group of MIT students led by James Harvey, a group called the Dr. John Lottery Club from Northeastern University, and a retired engineer named Jerry Selbee from Michigan.

How did the state of Massachusetts respond to the exploitation of the 'roll down rule'?

The state eventually became aware of the situation but did not act immediately because the high-volume players were generating significant revenue for the state. The situation only changed when it was publicly exposed by the Boston Globe.

What is the mathematical concept of 'projective geometry' mentioned in the transcript?

Projective geometry is a form of geometry in which there are no parallel lines; any two lines will always intersect at some point. It is an extension of Euclidean geometry and was initially developed to help painters depict three-dimensional scenes on a two-dimensional canvas.

What is the Fano plane and how does it relate to the lottery discussion in the transcript?

The Fano plane is a geometric configuration that represents a projective plane with seven points and seven lines, where each line contains three points and each point is on three lines. It is related to the lottery discussion as it provides a way to select lottery tickets that minimizes risk by ensuring a consistent return regardless of the winning numbers drawn.

What is the significance of the paper by RHF Denniston mentioned in the transcript?

The paper by RHF Denniston is significant because it provides a method for generating combinatorial designs that can be used to select lottery tickets in a way that reduces risk. The speaker used a similar approach to create a configuration for the Cash Windfall lottery that guarantees winning at least some of the top prizes.