# Mathematical Thinking: Crash Course Statistics #2

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.

42 😀

The fact that people are so passionate about political implications of the use of a certain system of measurement is actually making me lose faith in humanity.

Is there any web site where we can look for statistcs like the ones in the video? Like the chance to die by a bee bite and much much more.

Excellent!!

40,230 FT = 12.26 KM. There you go, stop complaining about it not being in metric. Easy way to think about 1 mile = to about a one and a half km, its closer to 1.6 but its a bit easier to think 1.5 for easy multiplication.

Ahhh! How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenburg

(if anybody liked her example at the end, fun stuff)

OMG the WW2 example just made my brain 100000 times smarter

i love the fact you said that we should illuminate our questions by numbers , in state of taking care of numbers . so fascinating

Stay out of sand holes

WHAT

THERE ARE MORE STARS THAN INSECTS??!?

I love this crash course thank you for the education!!

Sorry I crashed first a little dinner !!!

Thanbget myself a best way to get better !!!

Thinking about Darwin about the improbable way of his things being possible !!!

There is one in 7,600,000,000 chance that i will get a girlfriend.

<3

20,000,000,000,000 ÷ 323,000,000=61,919.50464643962 //LOL

nice chanel

Folks, please realise this is made by people from the US for people from the US. And you're already on the internet – if you don't know the conversion factor (~3ft/m, 1.6km/mile) then google it! Lazy ingrates.

She's kind of boring. I like the dude's fast pace way of doing it more.

5:12 Damn, all these years I've been playing 4 8 15 16 23 42. So close.

wait can 6th graders watch this

I'm late obviously but I really like this series?

came hwere just to tel to Use metric pleaaaaaaaaaaaaaaseeeeeeeeeeeeee

what is stat and now we describe the numbers

The old-timey newsflash was chuckle-worthy. Nice work on the accent!h

Am i the only one who thinks the intro song is way to loud?

I don't know how much 40.230 ft is until you say it's 7.5 miles…

And that my friends it why nearly everyone else in the world uses the metric system :p

I can see many commentators struggling with use of imperial system. Here in our Universities in Pakistan, we get to study both and our exams consists of half of the questions with data in imperial units and half of it with metric system. The struggle is real :/

can you try to make fun videos?

It honestly would not be so much effort to add the metric in brackets. Though I know the imperial system I personally find it too messy for practical use.

Here's a revolutionary idea – the time you spent complaining in the comments could have been put toward converting the figures to metrics yourself using Google

Loving this series!!

Nice to know my chances of winning the lottery are the same as dying from ebola.

Can you use the metric system?

Don't like her using the imperial units? Watch someone else's videos. You aren't paying her for this. Bye.

Ah what a time to be an American familiar with imperial systems

I love it when people say 'the lottery is stupid'… What other way do the vast majority of people have to become a multi-millionaire.

♥♥♥loved it!

Thank you crash course ❤️

Ok you're good. I'm on my 11 yrs old sons phone btw.

You are look alike of Maggie gyllenhaal…

شكرا شوان حميد ❤️

The sand hole example.. is dark.

This reminds me of the Chinese fortune cookie problem. In Canada they have a lotto called 6-49. Or in combinatronics 49 choose 6. Or 13.9 million combinations

In many Chinese restaurants they give out fortune cookies with your bill. People would use these numbers for picking lotto numbers. Of course there was no better chance to win then using any other 6 numbers then the 6 that was on the fortune cookie. What people did not realize was the printer of the fortune cookies was lazy and they did not have a random number generator for the six numbers. So the printer printed a few thousand fortune cookies using the same 6 numbers. So purely by chance of course after many years the lotto number matching the fortune cookie came up. The lotto commission discovered many people won using the same 6 numbers and people were disappointed to share the jackpot with many others. At first the commission suspected fraud but soon found out that people were using the numbers on the fortune cookie. The commission talked to the printer and asked them to use more random numbers when printing fortune cookies. So the advise in Canada is to NOT use numbers from fortune cookies because it decrease your chance of sharing the jackpot.

Adriane is a MILF!

People are complaining about her not using the metric system. ?

I run crosscountry, I understand miles perfectly fine. Maybe y'all should run a few to get the feel.

i've only scrolled for a minute in the comment section and i can say with confidence that the statistics is that 99% of the comments are about no metric system

Sho I bak a cak? I thi I sho.

"use a more familiar unit"

uses imperial system1:1 likelihood of commenters using metric system.

Everyone is complaining about just one thing! Calm down!

Terrorism is not statistical. If they have sleeper cells and they attack after 5 year but at the same time in order to maximize chaos and hide their tracks, it's not statistics. It's the avoidance of statistics by statistical minds.

i'm not into numbers at all like i tend to just zone out when i hear numbers just thrown into a sentence and then i get confused or stressed (obviously i cant stand math) but…. this video was definitely interesting and i did not zone out for one second. super informative and helpful! i'm currently struggling in a class teaching SPSS for Research Methods and i didn't have much background in stats (i forgot about it honestly) so thank you so much! will continue watching this series 🙂

Hi! I’m a comment not arguing about metric units over imperial, have a good day!

Real Madrid game??? They are nothing without Cristiano, they lost to Ajax, what else bro?!!

What a great series! So well written and produced. Excellent graphics as well

While I realize that I'm a year late, and I was grateful to find many people requesting the usage of Metric system, when I see the people defending the (exclusive) use of the Imperial System in the video, I wish to point out that the point of the lecture was the 'Visualization of Numbers'! When they're using a system that most of the world does not use, it becomes harder to get the point across. While I get the argument that the video was made in a region where the usage of imperial system is more prevalent and also the other argument where units are arbitrary in mathematics, the entire point of the video is visualization of mathematics and numbers, so the things that would help with it, should include the use of the Metric System.

I see that minecraft creeperThose complaining about the lack of a metric system…just use a converter on your computer. You’re here to learn, no?

Natalism will solve US debt

Terrific ..

why tf do I get ecosia adverts when I already use it… I thought advertising was smart…

8:17 *venomous snakes are venomous, not poisonous.

2019 AP Stats test?

please say your decimal numbers properly

Are those death odds descriptive or inferential?

bruh this is straight fire

IM BULGARIAN

Stay out of sand holes!

STAY OUT OF SAND JOLES!!!!

there will be massive medical breakthroughs, aubrey degrey is working on it plus others are too, and he says if you are under 40 you have good chance of living to 1000+ years

sounds crazy, but the single biggest thing that can make sure something happens, is if alot smart people with good funding are working on it..and they are

Im illaterate

when u got a final exam tomorrow and ur watching this so u don't fail

Can you please not reference a Real Madrid game as I have no intuition of the capacity of their stadium. Also, NEVER use the Metric system so that the rest of the world will understand how use the great American company GOOGLE to convert our imperial units to metric units. (All this is said 'tongue-in-cheek') uh oh better explain — refers to a humorous or sarcastic statement expressed in a mock serious manner.

9:47 she got me there …….?

Math and physic are two gathering elements that will change the world appearances as soon as possible the next generation is going to see more magical stuffs on the future is due physic and math .so by the way for all parents across the world I wish you all to teach your own kids free thinking rich of unlimited imagination to move forward to better life .

we need crash course math

REALLY need

Funny how apparently Europeans can't be bothered to look up the conversion from Imperial to Metric but expect Americans to know or learn metric.

I love your course but the case with the Bulgarian lottery is much more likely to have been a fraud, given the number of times the lottery has been accused of fraudulent activities.

Crash courses are really really great. I watch them with my kids in Pakistan. The only PROBLEM. everyone speaks super fast and it becomes difficult to process information. Please add a speed control button like in most YouTube videos. Many thanks.

hey crashcourse, your video production is really good. it really makes us think and imagine better. applause????

haha amazing with the bullets. i got bambozelled

Yes, yes I am watching this and I'm over a billion seconds old. Thanks for zing! LOL Didn't hurt AT ALL. XD

There are very few snakes in the world that are poisonous. Almost every snake that you need to be scared of uses venom to kill. Venoms can only kill you if it enters the blood. Poisons can kill through inhalation, ingestion, entering the blood, or sometimes passing through the skin. It applies to spiders and other creatures too. I don't mean to sound like a know-it-all but when people bring up this misconception, it infuriates me!

im taking this next semester, and I’m watching this just to get an edge Lol

Too fast to comprehend and retain. Too much crash and dash.

7:48

Leading the way, giving knowledge to the people! Awesome stuff!

8:38 i heard “ Abraham wald and the missing butt holes”

I submitted Korean subtitles. It's awkward, but please review it and add it.

I wish we would stop referring to American soldiers as heroes and romanticizing wars. Also saying American lives were saved while ignoring that improving the jets probably meant they were able to kill more people is very insensitive. The lives of your fellow nationals are not inherently more valuable that the lives of others around the world. This + the use of imperial system it is very clear who your target audience is. I wish you realized you are having an impact much larger than that, and these details are making foreign viewers feel alienated! Still, I love and recommend your work, Crash Course!

There is small request ,you have posted engineering and statistics but not mathematics. Please make mathematics videos for mathematics lovers

Thanks for your efforts

Giving this a chance. But v e r y slow start to static’s here.

My head hurts

There's John Green, then Hank Green and then Adrienne Hill.

I am sorry. I know this is a science and education channel in youtube and in my best channels but ı can't hold myself ı should tell that. In 1:11 we all can see a terrorist attack clearly but do we know where is this place ? I can say that this place in Turkey we have lived many terror attack and we have to guard ourself.

2:42 Thanks Obama

I feel like I understand Spock much better after watching this…

Did you know that most people will believe any statistics without proper understanding and research.

the subject is more like statistical thinking more than methematical thinking