Will your name become extinct? - Numberphile

[@korylemon8911]

I can confirm that the Lemon name has not died out.

[@Murgablodazor69]

If you bleed do you bleed lemonade? Im sorry

[@ceruchi2084]

LOL

[@chadd990]

* insert lemonparty joke here *

[@skrounst]

If the world gives you Lemons, I hope they are sons.

[@Glendragon]

yet

[@KubeSquared]

As an only son of an only son this video triggered my existential dread.

[@agfd5659]

You better have at least 10 kids

[@magichands135]

Time to get to it

[@paulgoogol2652]

time to step up that baby game

[@teddycouch9306]

@Joshua Grahm I mean.... it is irrational

[@SharayaYT]

Well i am the only daughter so my name will die with me in my branch lol

[@jibran8410]

How about we add a " -1 " which kills the brother next to you. That ought to spice things up.

[@oz_jones]

Jibran they are royality, afrer all

[@LaGuerre19]

I didn't think there was anything out there that could "spice up" a Lemon Party, but you found it! Hurrah!

[@z-beeblebrox]

Ah yes, the Cain variable

[@Wild4lon]

@@LaGuerre19 'hurrah' xD

[@hpekristiansen]

In my family the second son always kills an uncle.

[@ayush.kumar.13907]

"13 sons, all boys." What a Genius, I would have never guessed!

[@KL_Stereo]

DID YOU JUST ASSUME etc

[@ILOVEBIGCHUNGUS]

It's emphasis not explanation

[@matteo-ciaramitaro]

@Carey Hunt You mean 2 of the sons were boys too

[@ayush.kumar.13907]

@@matteo-ciaramitaro you didn't get the sarcasm there

[@matteo-ciaramitaro]

@@ayush.kumar.13907 i was going a level deeper. You missed the second level.

[@warhawk3972]

2 James Grime videos in one day? I'm in heaven.

[@James-zs3vm]

3

[@General12th]

@@James-zs3vm Double Heaven.

[@wobblysauce]

Seems like a scam.

[@pudicio4895]

We are unworthy

[@andrewlouie2]

3 if you check out standupmaths

[@qwertyuoip1234]

“17’s a number” Indeed.

[@ceruchi2084]

Doesn't he say "the number" as in, "the number of kids"?

[@wilwumn392]

Lol this has 17 likes. No offence but I’m not changing that.

[@titwarbler]

* rolls a zero for the first generation. game ends *

[@leftylizard9085]

I've played the game myself. Even in just the first incarnation of the game wherein there are 3 0's, a 1, a 2, and a 3, I still either end up losing the very first round or end up with too many circles and not enough pieces of paper to avoid taking out an entire rainforest.

[@themobiusfunction]

No zeros: *game never ends*

[@Triantalex]

??.

[@courtney-ray]

The entire video I was thinking about the scenario he called the one exception. So I’m glad he said it!

[@roy2615]

I thought about it a minute or so before he said it LOL

[@dijek5511]

The proof wouldn't help with that case either since x = G(x) would just be x = x which provides the restriction on x

[@mallardbro7487]

I read your comment before the video, forgot about your comment, and then went through the majority of the video thinking about the exception. I'm also glad he spoke about the exception.

[@MeneerBobRoos]

@@dijek5511 The proof still works out. You take x=G(x), which is true for all x=>0, you take the smallest x (as before) so the probability is 0.

[@MrDannyDetail]

@@MeneerBobRoos On the line x=x wouldn't the smallest place it touched be negative infinity? Then again a negative probability might not have much meaning.

[@nymalous3428]

I did something kind of like this once not too long ago when I was doing some personal research for a campaign of a roleplaying game that I was running. I wanted to have details of all of the families in some of the towns I had created for the game. I had a couple dozen family names for each town, and I worked up some rules for intermarrying and having kids (resolved by rolling dice). I noticed that eventually most of the family names went "extinct." It was not because they did not have any children, but none of their children were male (and in these fictional towns only males carried on the family name). So it didn't matter that every couple had children, what mattered was whether the children were male or female. I didn't even have to go through that many generations to have a large percentage of the names wiped out, maybe four or five. (And, no, I never did use the research or the fictional towns, the campaign fell through, people were too busy. I continued to do the math for a few weeks anyway, just because it was interesting, but eventually I had to give it up... the record keeping of thousands of individuals and their offspring was too tedious.)

[@kailomonkey]

The maths and spreadsheets/notes are always way more interesting than life :)

[@sarahchicago]

Perhaps that's why arranged marriages were and are so common in some societies. Much easier to ensure continuation of a family name with planning than with dice rolls.

[@nymalous3428]

SarahChicago Arranging a marriage is only a part of the process. The rest is up to nature, such as whether any given child will be male or female, or even whether the married couple have any children at all. The biggest problem I ran into during my little "experiment" were couples that only had female children; since the family name was carried by the male children, this occasionally resulted in a family name not being passed on. But I can see how arranging a marriage could at least help.

[@pussinboots9983]

This is interesting, especially in world building. I knew I should consider the family names.

[@josephlombardo1246]

I'm surprised the Dr. Grime didn't fully explain the intuition behind his equation. That is the essence of the problem, and much more interesting than the graphical solution to the problem. He should start with x is the probability that a line goes extinct starting with one individual. There is a 1/6 chance it ends right away, 1/3 chance there is one son and that son's line will go extinct with the same probability x, 1/3 chance there are two sons each with probability x of their line going extinct so probability of both going extinct is x^2, and 1/6 chance of 3 sons, each with probability x of their line going extinct so probability of all three going extinct is x^3. Therefore x=1/6+x/3+x^2/3+x^3/6. The equation seemed to come out of nowhere here.

[@Maazin5]

Joseph Lombardo thanks for clarifying. I had a hard time understanding that. Still don’t get why P(extinct by gen n) = G(P(extinct by gen n-1))

[@Stigvandr]

@@Maazin5 Yeah, I didn't get that either.

[@Joseduarte4867]

I agree i believe the critical part of this problem was that equation. Thanks for clarifying.

[@darrenr3712]

Also is x^2 and x^3 because the probabilities are assumed to be independent of each other.

[@adamsitabkhan5479]

Thank you!

[@anonanon9634]

Vsauce: will we run out of names? Numberphilie: will your name be extinct? I feel lost. What should I do

[@Rekko82]

Start a new game! :)

[@CosmiaNebula]

Do the math! And that clears your mind.

[@Ragnarok540]

Just accept it.

[@dschledermann]

Have a lot of children..

[@aurelia8028]

steal extinct names for yourself. Then we'll never run out

[@joedeshon]

Dr. Grimes explains complicated stuff more elegantly than just about anybody. Double his salary!

[@NikozBG]

Have you ever heard of a channel named 2blue1brown?

[@bobus_mogus]

@@NikozBG *3

[@thebrahmnicboy]

@@NikozBG I am pretty sure it is called 3.415Blue2.718Brown

[@x3ICEx]

0*2=0

[@wishiwasabear]

RIP Sir Lemon and his noble family. His family name will be remembered.

[@Triantalex]

false.

[@elsvanwin6816]

2 videos with James! Love it.

[@adamcrowe8372]

The family of 13 boys is from my hometown. That’s pretty cool. They did a write up on the family around 2004. All of the brothers served in the military, several in WW2. Never expected to hear Sullivan, Illinois in a Numberphile video!

[@chuuuu1131]

This applies to every name except Nguyen, the last name that will never die.

[@ErikB605]

@@Peter-q1p7t Because in the beginning there was exactly 1 last name per person -_-

[@chuuuu1131]

@@jshlst Mohamed is a first name

[@sam4330]

@@Peter-q1p7t Yeah, 40% of Vietnamese people are close relatives, that must be the only logical explanation to them having the same surname.

[@lobstertexas]

Because they keep Nguyinnig. Eyy-o!

[@o4_]

Omg, there are like 12 people at my school which have that last name, including a teacher.

[@noidea2568]

Wow, I just finished watching the unlisted version and you uploaded it now.

[@ProfRonanMC]

"A family of thirteen sons, all boys" - well, there's a coincidence!

[@douggjoseph]

Actually it's not. "All boys" was a needful statement to pre-answer the question of how many girls they had.

[@Triantalex]

??.

[@MittGGG]

Another video with James? I'm digging it!

[@jh-ec7si]

I was thinking on the first one how lucky he was that he didn't have to awkwardly draw the lines because of how close he made the dots, then the second set lost all hope.

[@KillianDefaoite]

This video does a nice job of briefly explaining some very deep mathematics. Awesome!

[@EebstertheGreat]

You aren't guaranteed to go extinct, but the probability of extinction approaches 100% (and equals 100% after infinitely many generations). But even if its probability is 0%, it is still possible to have more kids every generation. That still exists in the sample space, unless every number on the die is 0.

[@Shit_I_Missed.]

It's not possible to have more kids if a zero is rolled, and if the birthrate doesn't over compensate for the 0s rolled, it's going extinct

[@anticorncob6]

I thought the same thing. He probably just said “guaranteed” because for practical purposes, there’s no point considering those possibilities. It’s literally less probable than any positive real number.

[@WarwickAllison]

Exactly - if the probably of extinction only "reaches" 100% "at" infinity then it's misleading to say "guaranteed". His all-1 die isn't really an exception either, it's just the limit case of something more interesting: the 0-0-0-1-2-3 die reaches 50% at n=1 (since the first roll has a 50% chance of hitting 0), but a 0-1-1-1-1-2 die takes longer to reach 50%. You could for example have a 1000 sided die with 1 on every side except for one side with 0 and one side with 2. Same total probability (1), but clearly it now takes a *lot* longer to get close to 100%. We can keep adding sides making it less and less likely, approaching the all-1 die as the number of sides approaches infinity.

[@EebstertheGreat]

@@WarwickAllison I think that die is still exceptional in the sense that for every other die with mean 1, the asymptotic probability of extinction is 1, but for this specific die with mean 1, the asymptotic probability of extinction is 0.

[@kailomonkey]

@@EebstertheGreat This may conjecture that given infinite generations the

[@crankstonshnord6591]

"Sir-Lee Temple" hahahahahaha nice

[@johnhooyer3101]

I only noticed once I saw this comment emphasizing it. Thanks.

[@alexandermcclure6185]

Had to read it out in my mind *facepalms*

[@sabriath]

I had a dream about a variant solution to this problem, and came to the conclusion that you have 100% chance of your name going extinct as long as there is a "0" outcome available on the roll. There is always the probability that you will have all zeroes, but never the probability that a new name is created, so the list is deteriorating. It's similar to the wondering ant on a rubber band being stretched.

[@JustOneAsbesto]

I've heard the Lemons' family reunions are quite the party.

[@cyboticIndustries]

nah - i went to one, it was rather a bitter affair

[@SJrad]

cyboticIndustries I had a sour experience

[@LordBhorak]

At least in that party, they REALLY take care of the elderly men... maybe because they're the ones that have made the name live on. :P

[@agfd5659]

Ive been to a lemon party too. It's... different than what I expected

[@umchoyka]

They party like it's 1699

[@SightFilms]

i've just studied Applied Statistics at college recently, and it was really nice seeing something i learned in this video (the expectancy of a variable according to the probabilities of the outcomes it can be). This is a random comment but it was just nice seeing something i learned actually used in real life haha

[@booksquid856]

In Puerto Rico (and I believe in many Latin countries) kids receive two last names-mom’s first last name and dad’s first last name, hyphen in between. My kids have a name from me AND from hubby. Kind of changes the game.

[@wanderingrandomer]

Imagine my surprise at seeing this video fresh in my feed having watched it 2 hours ago.

[@LuizBHMG]

12:26 such a lovely noise!

[@Lizardfiz12]

was not expecting that expression at 7:09 😂😂 made my night

[@ShaunCockerill]

The dice with all sides of one gives the average roll of 1, but it also gives the formula g(x) = x^2. It crosses the x=y line at both 0,0 and 1,1.

[@ashtondemarse]

This video is simplified calculus of infinite series. Just took my exam and didn’t expect to think of this again.

[@alex_evstyugov]

Oh well. From the title I thought this was actually going to be about your *name*. Which you might well share with a hundred thousand people rather than just your immediate family, and yet even so it might, and indeed likely will, go extinct. Unless it's Smith, Li, or Nguyen. I believe the fancy name for that is "the Galton-Watson process". Wasn't there a Numberphile video on that a couple years ago? I can't seem to be able to find it.

[@kimfriedrich9770]

James's smile of certain extinction made my day.

[@TheRumpletiltskin]

"a family of 13 sons, all boys" really? All the sons are boys... man, i was confused before you pointed that out.

[@menjolno]

If you never grow in average, and you always have chances to lose, you will eventually lose all.

[@ceruchi2084]

Thank you! They should have made this explicit in the video. Now I understand.

[@WFDConnor]

you can go even further and say that if you have chances to lose all, aka each son in a generation has zero sons, you'll eventually lose all.

[@trdi]

This is great. Interesting topic and very elegant solution.

[@coreyburton8]

Another video with James- this is better than Endgame!

[@singularity3724]

For those interested, you can find out more by searching for “Galton-Watson Branching Processes”.

[@MrMineHeads.]

Waiting for the Numberphile podcast with James Grime

[@EverettWilson]

Ohhh, I just realized -- this is the gambler's ruin!

[@abj136]

It's not only last names that die out. There's an amazing video out there charting and graphing the decline and fall of 'Bob's in competitive sports. In this case it's more a memetic decline than direct children.

[@ThisUserHasBeenCanceled]

This was very unclear to me. Why multiply the probability by x^1, x^2 etc.? What does that mean?

[@Bdoserror]

He covers that in the extra bits video

[@VivekKumar-nh2dc]

3:42 that time lapse's audio is really scary

[@TemplerOO7]

Was about to write a smart comment about the dice with only ones on it and how it will never go extinct but he saw that one coming

[@PC_Simo]

Whenever I read / hear just the word: ”Name”, by itself; I always think of: ”First Name”, by default (this time, included); so, I’m kind of puzzled, why the title of this video didn’t have the word: ”Surname”, or the words: ”Family Name”, or something, like that 😮🤔😅.

[@LuisTorres-sq3xj]

A beaitoful lesson, everything in life is ruled by the mathematics ❤️.

[@johnplesia5154]

Reminescent of a Leslie's Model and Leslie Matrix. I imediately remembered that it depends on one special value (this case average) for the Matrix that either drives the population to extincion or lets it survive.

[@MyPisceanNature]

One really has to wonder why inheritance by the female line isn't preferred among monarchs. Up until quite recently, there is no question as to who the mother of a child is, genetically speaking. So, if you want to be sure your descendants inherit your assets, leaving those assets to the female lines seems most advantageous.

[@robson6285]

That man can explain maths on a way that is a joy to learn maths! Só great that he makes new numberphiles again!!

[@kelvinleung5065]

Gotta love that galton-watson process

[@Vermillionns]

What kind of maniac does not close an open parenthesis of an equation in a public video available on the internet...

[@michaelsommers2356]

Not a maniac, just someone whose mind is far above such trivialities.

[@Vermillionns]

@@michaelsommers2356 No, I'm pretty sure he wanted to hurt people.

[@ccmarcus]

I won't say, that you calculated the average. You calculated the expected value It's just my addition to this great video. :)

[@Empire526]

5:36 This is bringing me back to my biology test today(di-hybrid Punnit square)...

[@TheAstronomyDude]

How did the Victorians calculate the probabilities? How did they decide on the values for their dice?

[@TerribleTonyShow]

I played the dice game, and trust me, it is time consuming. Thank you Numberphile, for wasting my time preciously.

[@Valeriobrogni]

Sir Lemon at the top of the family tree ... I see what you did there

[@deamon6681]

Mind sharing your insight?

[@Valeriobrogni]

@@deamon6681 What?

[@MrMW2nd]

@@Valeriobrogni mind sharing your insight?

[@themonsterodub]

@@Valeriobrogni we don't see what he did lol

[@Valeriobrogni]

It's a lemon at the top of a tree, that's where lemons grow

[@AaronSmith1]

Dang it Numberphile, I'd just watched your last video and was about ready to get back to work and now this...

[@Yora21]

Sir Lemon? That's Earl Lemon-Grab! UNACCEPTABLE!!!!

[@ExtraterrestrialIntelligence]

make a video on family trees and acyclic graphs

[@ditrixgenesis781]

I thought, well certainly I could continue to roll numbers, even if the average is 1 or below. But then I remember the infinite forest episode, and think about how it's impossible to get a perfect integer by rolls, because eventually, you'll stray off of 0. So the fact that the chance

[@AlexFromTheWoods]

As a mathematician, I would like to point out the following: Although the probability of extinction is one in the case of a die with an average less or equal to one, the name could still survive if there's not just zeros on the die. Because if there is at least one 1 on the die, every man could have a son in each generation. The probability for that would be 0 in the limit case, but it would still be a possible outcome. It's actually this strange thing about infinite random experiment: Outcomes may have a chance of 1, and still they are not guaranteed.

[@nienke7713]

The 1 only die also has a different looking graph, as it would just be G(x) = 6/6 x = x, and as such it would just be equal to the dotted line (y=x) on every point, including 0, which automatically makes it the lowest non-negative number where G(x)=x and thus the probability of extinction will be 0, and any die which doesn't have a 0 on it will have a point G(X)=x at the origin and therefore never go extinct.

[@Danilego]

Two videos today and they're both related! They're also related to being related!

[@CheesyBread]

While watching this whole video i was so excited that I found an exception to his rule, until the very end of the video where he said exactly what I was thinking. :(

[@joshuarosen6242]

1:06 "...thirteen sons, all boys." what other sort of son is there?

[@MultiPenners]

A Fruit

[@sk8rdman]

in the 1860s my great-great-great-great grandfather came to America from Germany with his two sons. Each of them had around 5-10 kids, and each of their kids had 5-10 kids, and each of their kids had 5-10 kids, and so on. Needless to say, the family name is now very common where I live. I don't think we have to worry about the name dying out any time soon.

[@Scribblersys]

On the all-ones die, the resulting polynomial is the y = x line, and the first point in x = [0, 1] where it intersects itself is (0, 0), thus 0% probability of going extinct.

[@NikozBG]

Please James, next time actually roll the die, not just only drip it. My inner nerd was slowly dying during the first segment of this video.

[@gabeyk9]

13:45 I was thinking about that one THE WHOLE TIME

[@Amythebard2]

Ive finally found what probability generating functions are used for

[@shruggzdastr8-facedclown]

This from the Department Of Redundancy Department (@ 1:00): "...a family of thirteen sons -- all boys"! ;^}

[@bentoth9555]

I saw it as clarifying that he had 13 sons but no daughters.

[@shruggzdastr8-facedclown]

The audio said "thirteen sons" -- sons already implies boys and automatically eliminates daughters; so, adding "all boys" makes it redundant.

[@bentoth9555]

@@shruggzdastr8-facedclown Saying 13 sons says nothing on its own about the number of daughters he may have in addition. Saying "all boys" afterwards does, however.

[@michaelsommers2356]

@@bentoth9555 But the "all boys" qualifies the "thirteen sons"; it says nothing about any other children.

[@douggjoseph]

@@shruggzdastr8-facedclown : It's not redundant in the least. I have two sons. Nothing in that statement tells you how many daughters I have. Unless I add, "All boys" or "Only boys" or "No daughters" you've no idea that the total number of sons = total number of children.

[@MartyWoodcock]

My family line's surname ends with me. I have 2 sisters whom have changed surnames. I have 2 daughters and no sons. Chances are, my daughters won't keep their surname when (and if) they get hitched, leaving me as the last to bear my family name. None of my cousins have that family name. Making me, the last of my "kind".

[@pussinboots9983]

Or do you? Find your great grandfathers, then.

[@md7556]

Woah we just had this in school, so this is the first video where i actually know what he will do

[@jaredcarter1165]

2:57 That's actually pretty lucky for an example for the video that the Lemon family stopped at 4 generations; I experimented with this die myself and only went extinct after 34 generations.

[@rlamacraft]

Jared Carter maybe they did multiple takes, throwing away any that didn’t die off pretty quickly

[@Quick_Castyyy]

@@rlamacraft they cut the vid at 2:23 "to get a 0"

[@HeavyboxesDIYMaster]

I used the same dice to determine how many kids to have.

[@invisibledave]

I tried this but unfortunately all sides of the die had a "0" on it.

[@antonfalu123]

It's not guaranteed to die out, but the probability is 1, which is different. It is not impossible that the name will die out, but the set of such sequences has zero measure. James knows this of course, but it's a fun and subtle difference!

[@motivsmaras]

7:13 the best part ever, he is so happy 7:13

[@digthewarmth]

13:51 The moral of the story is make sure everyone in each generation has at least 1 son. That's why royals are obsessed with having a son.

[@ceruchi2084]

Poor Henry VIII couldn't manage.

[@mikasa3427]

I would like the $4.83/£ exchange rate mentioned at 5:00.

[@MaksiZockt]

the single one exception makes very much sense actually.. if the chance of no children is 0 then the Graph of G(x) touches the y = x line twice.. once at (0|0) and once at (1|1), really neat trick converting the probabilities to a polynomial

[@effuah]

This not only touches twice, it touches everywhere

[@davidgillies620]

Generating functions of dice are fun. Maybe Numberphile could do a video on Sicherman dice. The Wikipedia proof using cyclotomic polynomials is dear to my heart (I wrote it).

[@batfan1939]

I just watched two or three minutes of the sofa moving problem because I hit the next button on this video without realizing it. I thought it was a flashback, and was wondering why it ran so long DX

[@orange9089]

8:07 As soon as I heard co-efficients I got flashbacks to my A levels and I could feel where this was going

[@StichyWichy21]

Further Statistics. Probability generating functions.

[@orange9089]

Stich21 Yeah boi

[@stormysamreen7062]

Me: *cheats on exam Me: *fails Me: 2:21

[@aaronbeans33]

3 surnames in my family lines died out Bell, Fletcher & Chidley.

[@mrnicomedes]

OH NO. That's the probability of penultimate extinction! I really really hope he says those words in that order.

[@Ragatokk]

You need to explain why there is not a probability that you roll greater than 1 on average with a die that has 1 as the average. I would have assumed since it is random you could get an infinite amount of 2's rolled in a row even on a dice with less than 1 as the average IE 0-0-0-1-2, however unlikely it seems possible. I would assume the probability would be trending towards 0, but it seems like it would never hit 0.

[@innocent6083]

Cant you get extremely lucky and just roll 1s Infinitly many times even though your probability of doing it is extremely low

[@Joseduarte4867]

You can get lucky by rolling arbitrarely large Number of 1s but not infinitly many. Thats the definition of probability

[@Edgelordess]

I love math but why do I have to find all these interesting math videos right before I go to bed?

[@eXCeL2523]

Just learnt this from Stochastic Processes

[@toadounetlovesyou]

No comments before today? Was this video unlisted by mistake?

[@numberphile]

No, it was uploaded a while ago but was a secret - it'll be formally published later today, but you are seeing it early because you're a loyal viewer who clicked the links from the connected royal baby video!

[@questimegaming]

@@numberphile No royal baby videos here. Just a loyal viewer :D

[@stormysamreen7062]

The whole video is just Simon doing family planning

[@mathmachine4266]

I did a calculation for this awhile back. I wanted to know, if you start out with one sapling in Minecraft, what is the probability you run out of trees at some point? I came up with 7% ish. Too bad I forgot how I came up with that. EDIT: no wait, I remember now. Assuming P(0) is the probability that someone will have no children (drop no saplings), P(1) is the probability that someone will have 1 child (drop 1 sapling), P(2) is the probability that someone will have 2 children (drop 2 saplings), etc etc. Now you just have to solve for X, given that P(0)+P(1)X+P(2)X²+P(3)X³+P(4)X⁴+...=X. It's a polynomial equation. If there's currently only one person in the family (one tree), the probability of it dying out is X. If there are 2, the probability is X². If there are 3, the probability is X³, and so on. I'm pretty sure there are loads of special cases, but the only one I was able to find was when there's a 100% probability of having exactly 1 offspring, in which case the probability is undefined. Which, I mean...if you think about it, makes a lot of sense. Either it is or it isn't extinct in that case. EDIT EDIT: I watched the rest of the video. Hooray I was right!

[@StefanReich]

I don't see why you introduce the exponents in the first place. And what is x? 7:38

[@mr.sweetheart7507]

I have both my dad and mom's family names. Same for my brother and sister.

[@SheezyBites]

I don't understand how anything can fail to be guaranteed if it has any 0s. Certainly the chances on any individual step would be small, astronomically so in most high number cases, but extending to infinity it would have to happen eventually because it will never stop iterating. The chance of rolling all 0s is never itself 0, so surely in an infinite number of attempts it must eventually happen.

[@DS-xh9fd]

You should have mentioned this fun fact: if the average is exactly 1, then although it eventually dies out with 100% probability, the expected time to die out is infinite!

[@walterrobinson9796]

2 videos in one day?

[@ckq]

5:00 I mean basically the geometric mean is always less than the arithmetic mean