Mathematical Thinking: Crash Course Statistics #2

Summary

In this Crash Course Statistics episode, Adriene Hill explores the importance of numeracy in understanding and interpreting large and small numbers, which are often difficult for our intuition to grasp. She discusses strategies to contextualize big numbers, such as comparing the US debt to its population or converting large figures into more relatable units. Hill also touches on the 'law of truly large numbers,' illustrating how unlikely events become likely with large samples, using the Bulgarian lottery as an example. Additionally, she highlights the challenges of comprehending small probabilities, like winning the lottery or dying from rare causes, and how this can lead to misplaced fears. The video concludes with a historical example of mathematical thinking saving lives during WWII, emphasizing the power of numbers to inform decisions beyond just gut feelings.

Takeaways

• 🧮 Numeracy is crucial for understanding and interpreting data in everyday life, including political and economic discussions.
• 🚀 Mathematical thinking allows us to see beyond intuition and gut feelings, which are often not reliable for dealing with numbers.
• 👶 Our ability to comprehend numbers has limits, especially as they become very large or very small, affecting our intuition about quantities.
• 🌐 To make sense of big numbers, we can use context, comparisons, and time to relate them to more familiar scales.
• 💵 The U.S. debt can be personalized by considering the amount each citizen would owe, illustrating the magnitude in a relatable way.
• 📏 Converting large numbers into more familiar units, like miles or words, helps in visualizing and understanding their scale.
• ⏱ Time can be a powerful tool for comprehending large numbers, such as how many seconds are in a million or a billion.
• 🔬 Scientific notation is useful for calculations but may not always aid in intuitively understanding the magnitude of numbers.
• 🎰 The 'law of truly large numbers' explains why unlikely events can become likely when the sample size is large enough, as seen in lotteries.
• 🏆 Small probabilities, like winning the lottery, are difficult to grasp and can lead to misjudging risks and worrying about unlikely events.
• ✈️ Abraham Wald's story during World War II demonstrates how mathematical thinking can lead to life-saving decisions by analyzing data differently.

Q & A

What is the main focus of the Crash Course Statistics video presented by Adriene Hill?

The main focus of the video is to discuss the importance of statistics and how to think about numbers, particularly very large and very small numbers, in order to make sense of them and apply mathematical thinking to real-world situations.

Why is numeracy important according to the video?

Numeracy is important because it enables individuals to understand and interpret numerical information, such as political budgets, health risks, and probabilities, which is crucial for making informed decisions and assessing the allocation of resources.

How does the video suggest we can better understand very large numbers?

The video suggests several methods to understand large numbers: visualizing the difference between smaller increments, putting numbers in context, converting them into more familiar units of measurement, using reference points, and using time as a scale.

What is an example of putting a large number in context as mentioned in the video?

An example given in the video is the US debt, which is around 20 trillion dollars. When divided by the US population of about 323 million, it means each person owes approximately $62,500.

How does the video explain the concept of 'law of truly large numbers'?

The 'law of truly large numbers' is explained as the idea that with a large enough group or sample, even unlikely events become probable. The video uses the example of the Bulgarian lottery drawing the same numbers twice within a week to illustrate this concept.

What is the significance of understanding very small probabilities as discussed in the video?

Understanding very small probabilities is significant because it helps individuals assess risks accurately and prioritize concerns and actions based on actual likelihoods, rather than on exaggerated fears or misconceptions.

How does the video use the example of the Mega Millions lottery to illustrate the difficulty of comprehending small probabilities?

The video uses the example of the Mega Millions lottery, where the probability of winning is one in 302.6 million, to show how difficult it is for people to grasp such small probabilities. It compares this to the number of seconds in over 9.5 years, highlighting the near impossibility of such an event.

What is the story of Abraham Wald and the missing bullet holes, and how does it relate to mathematical thinking?

The story of Abraham Wald and the missing bullet holes is an example of mathematical thinking applied to real-world problem-solving. Wald analyzed data from World War II and noticed that planes returning from combat had fewer bullet holes in their engines. He deduced that this was because planes with engine damage were not returning, and thus recommended adding armor to the engines, which saved lives.

How does the video suggest mathematical thinking can help in decision-making?

The video suggests that mathematical thinking can help in decision-making by allowing individuals to see past coincidences, judge risks more accurately, and understand broader relationships in the world, providing a more rational basis for decisions beyond intuition.

What is the video's stance on the use of scientific notation for understanding large numbers?

The video notes that while scientific notation is helpful for calculations with large numbers, it may not be as intuitive for understanding them without context, as exponents can be non-intuitive and misleading without proper interpretation.

Outlines

00:00

🔢 Understanding Large and Small Numbers

Adriene Hill introduces the topic of statistics and the importance of numeracy, which is the ability to understand and interpret numbers and their implications in everyday life. The video aims to help viewers think mathematically about numbers, especially very large and very small ones. It discusses how humans struggle with numbers beyond their intuitive grasp and offers strategies to contextualize and visualize such numbers. Examples include comparing the US debt to the population or converting large numbers into more relatable units of measurement. The segment also touches on scientific notation and its limitations in conveying the true scale of numbers without context.

05:00

🎰 The Law of Truly Large Numbers and Risk Assessment

This section delves into the 'law of truly large numbers,' illustrating how with a large enough sample size, even unlikely events become probable. It uses the example of the Bulgarian lottery where the same set of numbers was drawn twice in a week, causing public outcry but being mathematically unsurprising. The discussion then shifts to the perception of risk and probability, particularly in the context of very low-probability events like winning the lottery or dying from Ebola. It highlights the importance of understanding small numbers to assess risks accurately and to prioritize concerns and actions effectively.

10:02

🛡 Mathematical Thinking in Decision Making

The final paragraph emphasizes the practical applications of mathematical thinking in decision-making. It recounts the story of Abraham Wald, who during World War II, used statistical analysis to advise the military on where to place armor on aircraft to maximize pilot survival. Wald's insight that the planes returning with fewer engine hits were not representative of all planes led to a strategy that saved lives. The paragraph concludes by underscoring the value of mathematical thinking in seeing beyond coincidences, assessing risks, and understanding broader relationships in the world, ultimately leading to better-informed decisions.

Keywords

💡Numeracy

Numeracy refers to the ability to understand and work with numbers and use them to solve problems in everyday life. In the video, numeracy is emphasized as a crucial skill for interpreting statistical data and making informed decisions. It's not just about performing calculations but also about comprehending the significance of numbers in various contexts, such as political discourse or risk assessment.

💡Intuition

Intuition is the ability to understand or know something immediately, without the need for conscious reasoning. The video discusses how intuition can often be misleading, especially when dealing with numbers and statistics. It suggests that mathematical thinking can help overcome the limitations of intuition, providing a more accurate and rational approach to understanding complex data.

💡Scientific Notation

Scientific notation is a way of expressing very large or very small numbers and is commonly used in scientific and mathematical contexts. The video points out that while scientific notation is useful for calculations, it can be non-intuitive for understanding the magnitude of numbers without proper context. It illustrates the importance of translating scientific notation into more relatable terms to grasp the true scale of quantities.

💡Law of Truly Large Numbers

The law of truly large numbers is a statistical concept that suggests as the sample size increases, the actual ratio of outcomes will converge on the expected ratio. The video uses this concept to explain the likelihood of coincidences and improbable events occurring within large populations, such as the repeated winning lottery numbers in Bulgaria, emphasizing that with a large enough sample, unlikely events become expected.

💡Risk Assessment

Risk assessment is the process of identifying, analyzing, and evaluating risks associated with a particular activity or decision. The video discusses how mathematical thinking can aid in risk assessment by providing a framework for comparing the probabilities of different events, such as the chances of dying from various causes. This helps in prioritizing actions and resources more effectively.

💡Statistical Thinking

Statistical thinking involves the application of statistical concepts and methods to analyze data and make inferences. The video emphasizes the importance of statistical thinking in interpreting data, making decisions, and understanding the world. It contrasts statistical thinking with gut feelings, highlighting the need for a more analytical approach to interpreting numerical information.

💡Probability

Probability is a measure of the likelihood that a particular event will occur. In the video, probability is used to discuss the chances of rare events, such as winning the lottery or dying from specific causes. It illustrates how understanding probabilities can help in making informed decisions and assessing the relative risks of different scenarios.

💡Contextualization

Contextualization is the process of placing information within a relevant context to make it more understandable. The video suggests that to comprehend large or small numbers, it's helpful to contextualize them within familiar frameworks, such as comparing the U.S. debt to the population or converting large numbers into more relatable units of measurement.

💡Reference Points

Reference points are specific, familiar quantities or events used to compare and understand new or unfamiliar data. The video uses reference points, such as the number of people attending a Dodgers game or the number of words in a novel, to help visualize and comprehend large numbers by relating them to more tangible experiences.

💡Coincidence

A coincidence is an event that occurs by chance and is not planned or intended. The video discusses how the law of truly large numbers can explain coincidences, such as the repeated lottery numbers in Bulgaria. It shows that with a large enough sample or number of trials, coincidences become statistically likely, challenging the perception of events as extraordinary.

💡Mathematical Thinking

Mathematical thinking is the process of applying mathematical concepts and techniques to analyze and solve problems. The video highlights mathematical thinking as a tool for seeing beyond intuition and making more informed decisions. It uses the example of Abraham Wald and the missing bullet holes to illustrate how mathematical thinking can lead to insights that save lives and improve outcomes.

Highlights

The importance of understanding statistics and mathematical thinking in daily life.

Numeracy is about comprehending numbers in context, not just calculation.

Mathematical thinking allows us to see beyond intuition and gut feelings.

Our ability to intuitively understand numbers diminishes as they become very large.

The challenge of visualizing large numbers like a million, billion, and trillion.

Using context to make sense of big numbers, such as comparing US debt to population.

Converting big numbers into more relatable units, like depth in miles.

Creating reference points for large numbers, like comparing crowds to known venues.

Using time to comprehend large numbers, such as a million seconds being less than 12 days.

The difference between scientific notation and understanding big numbers with context.

The 'law of truly large numbers' and its role in understanding coincidences.

The example of the Bulgarian lottery drawing the same numbers twice as a coincidence.

The difficulty of comprehending incredibly small probabilities, like lottery odds.

Comparing the likelihood of dying from Ebola to winning the lottery to put risks in perspective.

The importance of thinking through small numbers to assess what is worth worrying about.

The story of Abraham Wald and the missing bullet holes as an example of mathematical thinking.

How mathematical thinking can improve decision-making and risk assessment.

The conclusion that mathematical thinking provides insights beyond our intuition.

Transcripts

Hi, I’m Adriene Hill.

This is Crash Course Statistics.

In the last video we talked about why we care about statistics.

How we use statistics.

And Statistics is math.

So we thought we’d take a detour from the traditional curriculum to talk about how to

think about numbers.

Really, really big numbers.

Really small numbers.

And how to make sense of them.

We’re also going talk about mathematical thinking.

And fighter jets.

INTRO

Chances are, if you are watching this channel,

and certainly if you are commenting below, you are literate.

You understand language and how to use it.

But--are you equally comfortable with numbers?

I’m not talking about being able to calculate square roots in your head.

Or instantly tell whether or not 17321 is prime or not.

(It is.

I looked it up.)

Numeracy is about being able to wrap your head around what it means when politicians

talk about a one-point-five-trillion-dollar budget hole.

It’s about getting a handle on how much you should really lose sleep over the chance

of an Ebola outbreak.

And how to compare that risk to the chance of being killed by a terrorist.

Or a snake bite.

Or dying from an opioid overdose.

And what those comparisons might tell us about the time and resources we spend trying to

address those problems.

Mathematical thinking is about seeing the world in a different way.

Which means sometimes seeing beyond our intuition or gut feeling.

Because it turns out most of our guts are good at digesting food and pretty bad at math.

Infants less than a year old can discern between three objects.

I am much more advanced than an infant and can pretty easily comprehend the difference

in one and a hundred.

Even the difference between a hundred and a thousand or maybe even ten thousand.

For most of us, once numbers get really big, we lose our ability to have any intuitive

sense of them.

The distinction between a million and a billion and a trillion is really hard to visualize.

There are 100-trillion bacteria in each of our bodies and non-mathematical guts.

100-trillion.

There are an estimated 10- quintillion insects that are alive right now.

And an estimated 300-sextillion stars in the universe.

So how do you even begin to think about those big numbers and what they mean?

Let’s go to the THOUGHT BUBBLE.

Take a minute to visualize the difference between one and one hundred and one thousand

and a hundred thousand and a million.

That’s a lot of dots a whole lot of dots.`

There are other good ways to try to make sense of big numbers.

You can try to put the number in context.

The US debt is in the neighborhood of 20-Trillion dollars.

About 323 million people live in the US.

So--the debt owed for each person is about sixty-two thousand and five hundred dollars.

You can turn a big number into a unit of measurement you are more comfortable with.

The Kola Superdeep Borehole, which is the deepest artificial point on earth is 40,230 ft deep.

I have no idea how deep that is.

Until you tell me that it’s 7 and a half miles down.

And I can start to picture it.

You can have reference points for big numbers, ready to go.

There are about 100-thousand words in a 400 page novel.

About 46-thousand people show up to Dodgers games in Los Angeles.

I can roughly visualize that.

A million people taking to the streets to protest--might be easier to think of as 21

Dodger Stadium’s worth of people.

Or 14 and a half crowds for a Real Madrid match.

Time can help us go even bigger.

A million seconds is a little less than 12 days.

What about a billion seconds?

Do you think you are a billion seconds old?

Are you older than 32?

It takes 32 YEARS for a billion seconds to pass.

And what about a trillion seconds?

Think you or I will be alive after a trillion seconds passes?

Sorry to break it to you.

But no.

We will not.

Even if you are destined to be the Guinness Book of World Records oldest woman.

There is a 100% chance, barring massive medical breakthroughs that you will be dead.

It takes 32-thousand years for a trillion seconds to tick by.

Thanks thought bubble.

A quick note about scientific notation.

Scientific notation can be really helpful for calculating with big numbers, but not

necessarily helpful for understanding them.

Without context, exponents can be non-intuitive if that’s a word in their own way.

10 to the 39th and 10 to the 32nd sound like they might be close.

But 10 to the 39th is 10-MILLION times larger than 10 to the 32nd.

We’re not going to run the dots on that one.

There are about 7-point-six billion people on earth.

7-point-6 BILLION.

Understanding the sheer number of people out there in the world, can help us make sense

of the common-ness of coincidences or improbable events.

Some statisticians call it the “law of truly large numbers”.

The idea here is that with a large enough group, or sample, unlikely things are completely

likely to happen.

Consider this example from statistician David Hand.

On September 6th, 2009, the Bulgarian lottery randomly selected as the winning numbers

4, 15, 23, 24, 35, 42.

And then, four days later, on September 10th, the Bulgarian lottery randomly selected new

winning numbers.

4, 15, 23, 24, 35, and 42.

Exactly the same numbers.

People freaked out.

Bulgaria’s sports minister ordered an investigation.Was it fraud?

Something else?

Hand says calm down.

It’s just coincidence.

He lays out the math to prove it--but part of the argument here is the law of truly large numbers.

If you consider the number of lotto drawings--every week--around the world--over years and years--

“it would be amazing” he wrote, “if draws did not occasionally repeat.”

Speaking of those incredibly unlikely things...we need to talk about the flip side of incredibly

big numbers….

The incredibly small numbers--that can also be hard to comprehend.

Take the likelihood of winning a Mega Millions jackpot in the US.

Right now it’s about one in 302.6 million.

The probability that you’d win the jackpot is 0.000-000-003-305.

Let’s put the teeny-tiny chance of that happening in some perspective 302.6 million

is the number of seconds in more than 9 and a half years.

So, to borrow here from a very funny post by Tim Urban on Wait But Why that’s like

knowing that a hedgehog will sneeze once in the next 9 and a half years and betting on

the exact second... during those nine and a half years... that the hedgehog will need

a tissue.

I’m going with May 2nd, 2:23 and 33 seconds PM, 2021.

Our inability to judge small numbers does more than just cause us to misjudge our chances

of winning the lottery.

It causes us to worry about the wrong things.

To fear the wrong things.

Take your chance of dying from Ebola.

If you live in the US--the chance that you’ll be killed by Ebola in any given year is pretty

close to your chance of winning the mega millions lottery.

One in 309.6 million.

It is, among the very, very least likely ways anybody living in the US will die in a year.

Though you are, by some accounts, LESS likely to be killed by a terrorist attack in the

US committed by a refugee.

In a 2016 study, researchers calculated that at one in 3.64 Billion chance in the US in

a given year.

And you far more likely to be die in dozens of other ways.

There is a one in 6 million chance someone living in the US will be killed in a given

year by bee sting.

A one in 708-thousand chance that they’ll die falling from a ladder.

A one in 538 chance they’ll die in a given year from cancer.

And...not a big cause of death...but did you know people die in sand holes that they or

their friends have dug out at the beach.

They crawl in.

Looking for a little time in the sand hole.

And woosh.

The hole suddenly collapses and they are buried.

Stay out of sand holes.

I’m going to stop because it’s stressing me out.

The point here is that it’s worth taking the time to think through small numbers.

Cause they can help you figure out what’s actually worth worrying about.

And what isn’t.

What you might want to act on and what you might want society to take more and less seriously.

My personal take away is that I’d be way better off spending more time exercising and

less time looking around obsessively for poisonous snakes.

Cause the annual odds of dying from a snake bite in the US are only about one in 34

million, but the odds of dying from heart disease in any given year are one in 534.

Thinking mathematically isn’t just about understanding numbers better.

It’s about asking important questions about the world around us.

And letting numbers illuminate those questions.

One of my favorite examples mathematical thinking is the story of Abraham Wald and the missing

bullet holes.

Hat tip here to mathematician Jordan Ellenberg for highlighting the story.

Let’s go to the News Desk.

World War II: Manhattan.

A group of statisticians and mathematicians are hard at work trying to protect American

fighter pilots.

Their task--trying to figure out how to best armor planes without making them too heavy

so our heroes to outrun and outwit the enemy.

In an effort to figure out how to best protect our planes, the statisticians pour over data

of the planes that returned from fighting--looking at where they took damage.

Where the bullet holes were.

That data showed there were more bullet holes in the fuselage and fuel system and not as

many in the engines.

So how do we save our American heroes?

The exceptional statistician Abraham Wald studied the data and came back with the advice...that

surprised everyone to put the armor where the bullet holes weren’t.

Over the engines.

Wald realized the bullet holes should have been more evenly distributed over the planes.

If fewer planes were returning with holes in the engines--that meant those planes weren’t

returning home.

Wald has the exceptional realization the data wasn’t a random sample of all planes.

It only represented the planes that returned.

He suggested the military add armor to engines.

American lives were saved!

Not all mathematical thinking is going to help you save lives.

But it will help you make better decisions- Mathematical thinking can help you see past

coincidence.

It can help you judge risks.

It can help you see the broader relationships in the world.

Thinking mathematically gives us something to go on other than our guts, and their trillions of bacteria.

Thanks for watching. I'll see you again next time.