Why is this number everywhere?

Veritasium
28 Mar 202423:50

Summary

TLDRThe video script explores the intriguing tendency of people to choose the number 37 when asked for a random number between 1 and 100, a phenomenon known as the 'blue-seven phenomenon.' It delves into the psychological and mathematical reasons behind this pattern, highlighting 37's significance as a prime number and its prevalence in various aspects of life. The script also discusses the '37% rule' for decision-making, emphasizing the number's unexpected importance and widespread subconscious recognition.

Takeaways

  • 🎲 People tend to select numbers like 37 and 7 when asked to pick a random number, a phenomenon that has been observed across cultures.
  • 🌐 The 'blue-seven phenomenon' is a term used by psychologists to describe the pattern of people consistently choosing blue as a color and 7 as a number.
  • πŸ” A large survey conducted revealed that 37 and 73 are the most commonly chosen numbers when people are asked to select a random number between 1 and 100.
  • πŸ“Š The distribution of preferred random numbers shows a consistency that suggests a non-random pattern in human perception of randomness.
  • 🎩 Magic tricks, such as 'The 37 Force', rely on the predictability of people choosing the number 37.
  • πŸ“š The number 37 has a unique mathematical significance, being a prime number that stands out for its properties and its occurrence as a second prime factor.
  • πŸ€” The perception of randomness might be influenced by the rarity of prime numbers in our daily lives and the lack of a formula to predict them.
  • πŸ“ˆ The secretary problem, or the marriage problem, is a mathematical question that suggests an optimal strategy for making decisions, which happens to involve the number 37.
  • 🌟 The number 37 has a special place in human intuition and decision-making, often being chosen as a 'random' number in various contexts.
  • 🌍 The ubiquity of 37 in various aspects of life has led to a fascination and collection of instances where 37 appears, highlighting its prevalence and significance.
  • πŸš€ The number 37, through its mathematical properties and its role in decision-making, has captured human interest and curiosity, leading to a deeper exploration of its role in our lives.

Q & A

  • What is the 'blue-seven phenomenon'?

    -The 'blue-seven phenomenon' refers to the tendency of people across different cultures to consistently choose the color blue and the number 7 when asked to select something randomly.

  • Why do people often choose the number 37 as a random number?

    -People often choose the number 37 as a random number due to a psychological pattern, possibly because it feels random and is a two-digit prime number. Additionally, it has been suggested that 37 is the equivalent of the 'blue-seven phenomenon' for numbers between 1 and 100.

  • What is the significance of the number 37 in the context of magic tricks?

    -The number 37 is significant in magic tricks because there is a widespread professional magic trick called 'The 37 Force,' which relies on getting an audience member to seemingly choose 37 out of thin air.

  • What does the '37 Website' document?

    -The '37 Website' documents instances and occurrences of the number 37 in various contexts, as collected by a man who has been obsessively gathering these instances since the 1980s.

  • What is the 'secretary problem' or 'marriage problem'?

    -The 'secretary problem' or 'marriage problem' is a mathematical question about the optimal strategy for making a decision when faced with a series of options, such as hiring the best employee or choosing the best life partner.

  • What is the 37% rule in decision-making?

    -The 37% rule suggests that one should explore and reject 37% of options to get a sense of what's available, and then choose the first option that is better than all previous ones. This approach maximizes the chances of selecting the best option.

  • What is the median second prime factor of all numbers?

    -The median second prime factor of all numbers is 37, meaning that half of all numbers have a second prime factor that is 37 or less.

  • How do primes feel random to people?

    -Primes feel random to people because they don't appear as frequently in our daily lives and there is no formula to predict them; one can only check each number sequentially to find the next prime, making them feel more random than composite numbers.

  • Why do multiples of 10 not feel random to people?

    -Multiples of 10 do not feel random because they are symmetrical and central within the number range, which is considered too contrived or predictable compared to the irregularity and uniqueness of prime numbers like 37.

  • What is the significance of the number 37 in the context of the 'secretary problem'?

    -In the context of the 'secretary problem,' 37% of the available options should be explored and rejected to get a sense of the available choices, and then the first option that is better than all the previous ones should be selected, maximizing the chances of making the best decision.

  • How does the concept of randomness relate to prime numbers?

    -Prime numbers are often perceived as more random because they are less frequent in our daily experiences and lack a predictable pattern or formula, making them feel more irregular and unpredictable compared to composite numbers.

Outlines

00:00

🎲 The Curious Case of the Number 37

This paragraph delves into the peculiar tendency of people to choose the number 37 when asked to pick a random number between 1 and 100. It explores the 'blue-seven phenomenon' and how 37 has become a sort of default 'random' choice across cultures. The conversation highlights the surprising frequency with which 37 is selected in various surveys and experiments, leading to an investigation into the significance and prevalence of this number in everyday life and culture.

05:00

πŸ”’ Patterns in Randomness: The Prominence of 37

This section examines the patterns that emerge when people are asked to select random numbers, with a focus on the numbers 7, 37, and 73. It discusses the human perception of randomness and the subconscious attraction towards certain numbers, particularly odd ones like 37. The explanation extends to the mathematical properties of primes and their role in creating a sense of randomness. The paragraph also introduces the concept of the 'median second prime factor' and how 37 fits into this intriguing mathematical context.

10:00

🎩 Magic and Mathematics: The Enigma of 37

This part of the script uncovers more about the number 37, revealing it to be a prime number with many interesting properties, such as being an irregular, Cuban, lucky, sexy, permutable, and Padovan prime. It shares anecdotes of personal experiences and societal phenomena related to the number, including its appearance in literature and everyday life. The discussion also touches on a mathematical trick involving multiples of 37 and the number's significance in decision-making scenarios.

15:02

πŸ“š Life's Tipping Point: The 37% Rule

This paragraph introduces the secretary or marriage problem, which is a mathematical model for making optimal decisions when faced with a series of options. It explains how the number 37 plays a crucial role in this model, suggesting that one should reject the first 37% of options to understand the range and then choose the first option that is better than all previous ones. The discussion includes real-life applications of this strategy and its implications for major life decisions.

20:02

🌟 The Ubiquity of 37: A Collector's Passion

The final paragraph of the script tells the story of an individual's lifelong fascination and collection of instances and occurrences of the number 37. It highlights the sheer volume of 37-related items and anecdotes that one can find in the world, from everyday objects to significant life events. The conversation touches on the psychological and possibly universal draw of the number, its role in human intuition, and the impact of this shared fascination on personal pursuits and collections.

Mindmap

Keywords

πŸ’‘Randomness

Randomness refers to the quality of being unpredictable or not following a discernible pattern. In the context of the video, it is explored through people's attempts to choose random numbers, which often turn out to be not as random as one might expect. The video reveals that certain numbers, like 37, are chosen more frequently than others, demonstrating that human perception of randomness is not truly random.

πŸ’‘Blue-Seven Phenomenon

The Blue-Seven Phenomenon is a psychological observation that when people are asked to choose a color and a number, they often select blue and the number 7. This phenomenon highlights the predictable nature of human choices, which are thought to be random. The video extends this concept to the number 37 when choosing numbers between 1 and 100.

πŸ’‘Perception

Perception refers to the way humans interpret and understand sensory information to give meaning to the world around them. In the video, perception is discussed in relation to how people perceive the concept of randomness and how this affects their choices. The number 37 feels random to people, which is why it is often chosen despite its frequency of selection.

πŸ’‘Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. They are significant in mathematics due to their unique properties and occurrence. In the video, prime numbers are highlighted as feeling more random to humans, with 37 being a prime number that stands out as particularly interesting and commonly chosen.

πŸ’‘The Secretary Problem

The Secretary Problem, also known as the Marriage Problem, is a mathematical model that helps in making optimal decisions when faced with a series of choices, not knowing when the best option will appear. It suggests a strategy of rejecting a certain percentage of options initially to gain information, and then selecting the first option that is better than all previous ones.

πŸ’‘37 Website

The 37 Website is a dedicated online platform that the video's subject created in 1994 to collect and share instances of the number 37 appearing in various contexts. It serves as a hub for people from around the world to contribute their sightings of 37, showcasing the pervasiveness of this number in everyday life.

πŸ’‘Collecting

Collecting refers to the act of gathering and storing items or data of a particular type, often due to a personal interest or obsession. In the video, the subject has been collecting instances of the number 37 for many years, demonstrating a deep personal connection and fascination with the number.

πŸ’‘Intuition

Intuition is the ability to understand or sense something without the need for conscious reasoning. It is often considered a powerful tool that allows individuals to make decisions or come to insights based on subtle cues or feelings. The video suggests that the number 37 might hold a special place in human intuition, being chosen frequently as a 'random' number.

πŸ’‘Brilliant

Brilliant is an online learning platform that offers interactive lessons in a variety of subjects, including math, science, and technology. The platform aims to sharpen problem-solving skills and critical thinking by providing hands-on learning experiences. In the video, Brilliant is mentioned as a sponsor and is promoted as a resource for enhancing intuition and problem-solving abilities.

πŸ’‘Decision Making

Decision making is the process of selecting between alternatives, often involving weighing options against each other based on criteria and preferences. The video discusses decision making in the context of the Secretary Problem, highlighting the strategy of rejecting a percentage of options to maximize the chances of making the best choice.

πŸ’‘Pattern Recognition

Pattern recognition is the ability to identify regularities or repeating sequences in data or experiences. It is a cognitive process that allows humans to make sense of complex information by detecting underlying structures. In the video, pattern recognition is central to the discussion of why people repeatedly choose the number 37 as a random number, suggesting that there may be an unrecognized pattern in human behavior.

Highlights

People are often poor at selecting things randomly, with a tendency to choose certain numbers more frequently.

The 'blue-seven phenomenon' refers to the common pattern of people choosing blue as a color and 7 as a number.

The number 37 is often chosen as a 'random' number, leading to the investigation of its significance.

A large survey was conducted to test the theory that people often pick the number 37 at random.

The results of the survey showed that 37 was indeed one of the most commonly chosen numbers.

The number 37 has a unique mathematical property as a prime number, which might contribute to its perceived randomness.

Prime numbers are perceived as more random due to their infrequency in everyday life and the lack of a formula to predict them.

The median second prime factor of all numbers is 37, meaning half of all numbers have a second prime factor less than or equal to 37.

The number 37 has many special properties in mathematics, such as being an irregular, Cuban, lucky, sexy, permutable, and Padovan prime.

37 is used as an example in early childhood education to illustrate the concept of prime numbers.

The 37 rule is introduced as a strategy for making decisions, suggesting to explore and reject 37% of options before making a choice.

The 37 rule can be applied to various life decisions, such as choosing a job, renting an apartment, or even selecting a life partner.

A personal collection of instances where the number 37 appears in everyday life has been amassed over the years.

The fascination with the number 37 has led to the creation of a website dedicated to its occurrences.

The number 37 is deeply embedded in human intuition and perception of randomness, influencing our choices and decisions.

The video explores the cognitive and mathematical reasons behind the allure of the number 37.

The number 37 is so ingrained in human culture that it has become a part of various jokes, tricks, and even professional magic.

The video concludes by suggesting that the number 37 might be more than just a random choice, but a number that holds significance in various aspects of our lives.

Transcripts

00:00

- Let me show you something unbelievable.

00:02

Name a random number between 1 and 100.

00:05

- 61.

00:06

- Okay, that's pretty random.

00:07

- [Emily] Just name a random number from 1 to 100, random.

00:09

- 43. - 43, thank you so much.

00:11

- 56. - 7.

00:13

- I want the most random number between 1 and 100,

00:16

like totally random.

00:17

- 11.

00:18

- 37.

00:19

- [Interviewee] 79. - 79, thank you so much.

00:21

- 91. - 7.

00:23

- 3. - 37.

00:24

- [Derek] 37. - 37, yeah.

00:25

- [Derek] Why 37?

00:26

- I dunno, it's the first number that came to my mind.

00:29

- 44. - 27.

00:30

- 37. - 72.

00:32

- 4. - 13.

00:33

- 7. - 37.

00:34

- [Emily] Really?

00:35

(Derek speaking in foreign language)

00:39

- 13. - 7.

00:40

- 37. - 37?

00:41

- 73. - 37.

00:43

- 35. - 37.

00:44

- [Emily] 37, no way!

00:46

- 43. - 2.

00:47

- 37.

00:48

(Derek gasping)

00:49

- I knew you were gonna do it.

00:51

He just "37-ed" and walked away.

00:53

- Between 1 and 100.

00:54

- Ah, no thanks. - [Emily] Okay.

00:56

- 37.

00:57

- [Emily] Oh, perfect. Thank you so much.

00:58

- 83. - 37.

01:00

- 37.

01:01

- 87. - 55.

01:03

- 37. - 37.

01:04

(Emily gasping)

01:05

Can I shake your hand?

01:07

- [Derek] I love the thought you're putting into this.

01:08

- 37? - No, you are kidding me!

01:11

Are you real? - Yeah why?

01:13

- Did we ask you this already? - No.

01:15

- Random number between 1 to 100.

01:17

- 37. - 37. Oh, my gosh, yes.

01:20

- [Derek] Name a random number between 1 and 100.

01:22

- 37. - Are you kidding me?

01:26

Why?

01:28

- It's a good number, I guess, any number.

01:31

- [Derek] Where did that come from?

01:32

- Imagination, I suppose.

01:35

- So, what's going on?

01:37

Well, people are actually really bad

01:39

at selecting things randomly.

01:41

In fact, when asked to pick a color and a number,

01:43

people reliably select blue and 7 the most

01:47

across dozens of different cultures.

01:49

Psychologists have a name for this pattern.

01:51

The blue-seven phenomenon.

01:54

And when picking a random number between 1 and 100,

01:57

it has long been suggested

01:58

that the equivalent of the blue-seven phenomenon

02:01

is the number 37.

02:03

My producer, Emily, and I spoke to hundreds of people

02:05

to test this theory.

02:07

The most common answer was 7,

02:09

but maybe that's because people just expected

02:11

that we'd ask them for numbers between 1 and 10.

02:14

The most common two-digit number really was 37,

02:18

much to our surprise.

02:19

(Derek and Emily gasping)

02:25

So we decided to embark on the biggest investigation ever

02:29

on the number 37.

02:31

And it took us to some unexpected places.

02:34

- I think 37 is a fascinating number.

02:36

It's just really interesting because it turns up so much.

02:39

How many objects are there here in the room with us

02:42

that have a 37 on them?

02:43

I'm sure there's more than 1,000 here.

02:46

I built the 37 Website in 1994.

02:49

I started getting email from strangers, it's everywhere.

02:51

I'm trying to collect them all.

02:53

We're tireless.

02:54

The tireless cabal of 37 people, yeah.

02:58

- Apparently, people choose 37 so reliably

03:01

that there's even a widespread professional magic trick

03:04

that relies entirely on getting an audience member

03:06

to just pick 37 out of thin air.

03:09

It's called The 37 Force.

03:12

- I'm gonna ask you to think of a number in a moment, okay?

03:15

It's a two-digit number, less than 50.

03:18

Both numbers are odd, but different.

03:20

You could have 19, 17, or 15, but not 11.

03:22

Because you see both numbers are the same, 1 and 1 next to one another.

03:26

You ready?

03:27

One, two, and three.

03:29

What number did you think of?

03:31

- [Audience] 37. - 37.

03:32

Fascinating.

03:35

In the famous Stanford MIT Jargon File,

03:38

the origin of hacker slang,

03:40

37 is given as the random number of choice

03:43

for computer programmers.

03:44

β€œWhen groups of people are polled

03:46

to pick a random number between 1 and 100,

03:48

the most commonly chosen number is 37."

03:52

(graphic buzzing)

03:52

The thing is, no formal polls on this actually exist.

03:56

The best we found was a Reddit poll of 1,380 people

04:00

from four years ago,

04:02

and the most popular number was...

04:04

69.

04:06

But after that, the winning number was 37.

04:10

But we can do better than a sample size

04:11

of just 1,000 people.

04:13

So we conducted the largest random number survey ever.

04:17

In a community post 3 weeks ago,

04:18

we asked people to pick a random number between 1 and 100.

04:22

We received 200,000 responses.

04:26

Here are the results as they came in.

04:30

It's fascinating to watch

04:31

how consistent these supposedly random numbers are,

04:35

from 10,000, to 100,000,

04:39

all the way up to 200,000 respondents.

04:43

The distribution barely changes,

04:45

suggesting that people from all around the world

04:49

think about random numbers in a particular way,

04:52

and it is decidedly not random.

04:56

Ignoring the extremes of the scale

04:58

because people were primed

05:00

by the numbers 1 and 100 in the question itself,

05:02

and ignoring 42 and 69 because they're not random,

05:06

there are a few numbers that stand out,

05:09

which we seem to regard as more random than the rest.

05:13

7,

05:13

73,

05:15

77,

05:16

and 37.

05:17

(pensive music)

05:20

Then we asked people to pick the number

05:21

they thought the fewest others would pick.

05:24

The goal was to get rid of favorite or lucky numbers

05:26

and give truly random selections.

05:29

And here, the results were even clearer.

05:32

Again, ignoring the very extremes and 50 in the middle,

05:35

the most selected numbers were, far and away, 73 and 37,

05:40

which were nearly tied.

05:43

The actual least-picked number in the first question was 90,

05:47

followed by 30, 40, 70, 80, and 60.

05:50

Multiples of 10 apparently don't seem that random.

05:54

The most picked overall numbers ignoring the outliers

05:57

were 73 and 37.

05:59

(pensive music)

06:01

Ironically, all this evidence points to 37

06:04

and its inversion, 73, as not being random at all.

06:08

So why does everyone pick them?

06:10

Well, one argument

06:11

is that this is just how people perceive randomness.

06:14

37, does that feel random to you?

06:17

- Yeah. Yeah, it does.

06:19

- [Derek] Yeah, 50 wouldn't be random?

06:21

- No. - [Derek] No.

06:21

- It would be too contrived.

06:23

- [Derek] Yeah. - Yeah, it's too central

06:26

- I think people think that even numbers

06:28

are less random than odd numbers.

06:30

- 5 feels not random, 9 and 1 feel too extreme,

06:33

so people tend towards 3 and 7.

06:35

- This is backed up by the fact

06:36

that every one of the top numbers in our survey

06:39

consisted of 3s and 7s.

06:42

In fact, 3 and 7 were the most selected digits

06:45

on both questions.

06:49

But there's also a mathematical case

06:51

for humanity's number of choice

06:53

because it's not just odd numbers,

06:54

but specifically primes,

06:56

which feel like the most random numbers.

06:59

Notice how we ignore odds ending in 5s

07:02

or how something like 39

07:03

still feels a little less random than 37?

07:08

Primes feel random for at least two reasons.

07:11

First, they don't appear as much in our lives.

07:14

I mean, pixel counts, fruit boxes, square footage.

07:17

We live in a composite world with multiple dimensions

07:20

that multiply together,

07:22

so we just don't see primes much past the single digits.

07:26

Second, we don't have a formula for primes.

07:29

If you have a prime number

07:30

and you want to find the next one,

07:31

you have no choice but to check every number

07:34

until you find a prime.

07:35

The closest thing we have to a formula

07:37

is the prime number theorem,

07:39

which gives the approximation

07:40

that the nth prime number occurs around n

07:43

times natural log of n.

07:45

For example, the 1,000th prime number

07:47

should be around 6,908.

07:50

And it's close, but certainly not exact.

07:54

So primes essentially occur at random,

07:59

but of all the primes, 37 has reason to stand out.

08:02

(pensive music)

08:04

If we were to find the prime factors of every number,

08:08

we would see that 2 is the smallest prime factor

08:11

for exactly 1/2 of them,

08:13

all of the even numbers.

08:14

And 3 is the smallest prime factor

08:17

for 1/6 of all numbers,

08:18

anything that's divisible by 3 but not by 2 and so on.

08:22

As we pick larger and larger primes,

08:25

they form the smallest prime factor

08:26

for fewer and fewer integers.

08:29

But, what if we track the second smallest prime factor

08:32

of each number?

08:34

Well, first, we have 3,

08:35

which is the second prime factor of a number.

08:37

Only when the number is divisible by both 2 and 3

08:41

or divisible by 6.

08:42

So 1/6 of all numbers have a second prime factor of 3.

08:47

And as we keep going,

08:48

which number will end up at the balancing point?

08:51

This is the median second prime factor of all numbers,

08:55

all numbers from 1 all the way up to a googol

08:58

and off to infinity.

09:00

Would you believe that that number is 37?

09:03

(pensive music)

09:05

Let's take a look at 5.

09:06

5 is the second prime factor

09:08

only when a number is divisible by 5 and 3,

09:11

but not 2.

09:12

Or 5 and 2, but not 3.

09:15

In the first case, a number divisible by 5 and 3

09:18

means it's divisible by 15,

09:19

so that's 1/15 of all numbers.

09:21

But it also can't be divisible by 2.

09:24

So 1/2 of 1/15 is 1/30 of all numbers.

09:28

In the second case,

09:29

a number divisible by 5 and 2

09:30

means it's divisible by 10,

09:32

but it cannot be divisible by 3.

09:35

So we're left with 1/10 times 2/3

09:37

equals 1/15 of all numbers.

09:40

Adding up these two cases,

09:41

we get that 1/10 of all numbers

09:43

have 5 as their second prime factor.

09:46

And we can repeat this for the next prime, 7.

09:49

Just take each of these cases

09:51

and add them up to get that 1/15 of all integers

09:54

have a second prime factor of 7.

09:58

And so on.

09:59

Keeping a running total,

10:00

we quickly approach a balancing point

10:02

for the second prime factor across all integers.

10:05

And then we reach it.

10:08

So the median second prime factor of all numbers is 37.

10:13

Half of numbers have a second prime factor of 37 or less.

10:20

There are other remarkable qualities about 37 as a prime.

10:23

It's an irregular prime, a Cuban prime, a lucky prime,

10:27

a sexy prime, a permutable prime, a Padovan prime.

10:30

And at this point,

10:31

mathematicians might just be making up types of primes.

10:34

- 37's identity as a prime number is so strong

10:37

that the same day I first learned the number 37,

10:40

I learned it was prime.

10:42

This was one of my first books as a toddler.

10:44

It teaches you every number from 1 to 100

10:46

with a short story or fun fact for each.

10:49

So for 26, that's how many letters in the alphabet.

10:51

Or for 30, they give the days of September.

10:54

Or for 52, that's how many cards are in a deck.

10:57

Except 37.

10:58

(pages rustling) (jaunty music)

11:00

It's a prime number.

11:02

Nothing goes into it.

11:03

Someday, you'll understand.

11:06

I did not like that.

11:08

I understood every other number,

11:09

so I also wanted to understand 37.

11:11

So, that number has nagged me ever since,

11:13

and now this video is being made some 20 years later.

11:17

- [Derek] Not convinced yet?

11:20

- If you take a number that is a multiple of 37 already,

11:23

like 1, 3, 6, 9, that's 37 squared,

11:26

and then you reverse it,

11:28

and then you stick a 0 in between every digit,

11:32

then that number is a multiple of 37.

11:34

And I literally spent the next month on the bus

11:38

trying to prove that fact, which I finally did.

11:40

Just rattle off a six-digit number.

11:42

Tell me any six-digit number.

11:44

- 413,625.

11:48

- And it's not divisible by 37.

11:50

So how did I figure that out?

11:52

There's a trick for that.

11:54

- Is this your like party trick that you can bring out?

11:57

- Surprisingly,

11:57

it doesn't impress as many people as you would think.

12:00

I think it should impress everybody.

12:03

- But there's also a practical reason

12:04

37 is an important number for humanity.

12:07

Say you are faced with a choice

12:09

that is both immediate and final,

12:12

like whether to rent the apartment you've just toured

12:14

or whether to accept a job offer you received.

12:17

Or it can be as small

12:18

as whether to stop the next gas station on a road trip.

12:21

These are all problems

12:22

where you can't assess all the options at once

12:24

and then decide.

12:26

With each option you encounter,

12:28

you need to decide whether to accept it or reject it forever

12:31

and see what comes next.

12:33

In these scenarios,

12:34

it feels impossible to make the best choice.

12:37

If you select too early,

12:39

you'll probably never even see the best option.

12:41

But if you select too late,

12:42

well, then you've probably rejected the best option already.

12:46

So your best bet is somewhere in the middle.

12:49

There, you know at least some information

12:51

from the options you've seen,

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and you have some choice, to select or pass.

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But how do you know exactly when to decide?

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The optimal strategy looks like this.

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First, you need to see some options

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and reject them automatically

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just to learn what's out there.

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And then at a certain stopping point, S,

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you need to stop rejecting them

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and start evaluating whether an option

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is the best you've seen so far.

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If it is, then select it.

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But when should that stopping point be?

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We need to work out which stopping point

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maximizes our chances of picking the best option.

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We can calculate these chances.

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For each spot, find the probability

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that the best option is located there

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times the probability we get there from stopping point S.

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Then, add these probabilities up across every spot.

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Now, the chance of the best option being in any spot

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is just random.

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If there are N options in total, it's 1/N,

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but it's a little harder

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to find the chances of getting to each spot.

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Say the best option is in the next spot after S, S + 1.

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What are the chances we get there?

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Well, since this is the next spot over

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from the stopping point,

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we have 100% chance of getting there.

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So we are guaranteed to visit it and select it.

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But if the true best option is in spot S + 2,

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well, there's a small chance we'll miss it.

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If the best of all the previous options

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is sitting in spot S + 1,

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we would just pick that and stop looking

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before reaching S + 2.

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There's a one in S + 1 chance of this happening.

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So the chances we do get to spot S + 2

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to pick the true best option is 1 minus that,

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or S over S + 1.

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This same calculation continues up until the last spot N.

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We only get here

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if we've been passing on every option so far,

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which means that one of the first S options

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must have been the best

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of the total N - 1 options we've seen.

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In total, this gives us the expression 1/N

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times 1, + S over S + 1, plus S over S + 2, and so on,

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all the way up to S over N - 1.

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Factoring out the S,

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the sum inside the parentheses

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approximates the function 1/x going from S to N.

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(pensive music)

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Taking that integral, we get the natural log of N over S.

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So the probability we select the best option

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is S over N times the natural log of N over S.

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To maximize this probability,

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we can find the peak of this function

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by setting its derivative to 0,

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and this gives the natural log of S over N equals -1.

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So S over N equals 1 over e,

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or about 37%.

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So explore

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and reject 37% of options

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just to get a sense of what's out there,

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and then pick the first option to come along

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that's better than all of the ones you've seen so far.

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And your chances of success using this method

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are also 37%.

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(pensive music)

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This math question is known as the secretary problem

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or the marriage problem,

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as it also applies to hiring the best employees

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or even deciding on the best life partner.

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Now, it can be impractical to check 37% of the options

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because you don't always know

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how many candidates are out there,

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but the 37% rule also works for time.