@@ekinteko I believe you and your comment to be genius!!! 😱 I believe this based on my never having thought of this example nor have I seen anyone else explain bayesian statistics with such a sublimely simple example. I, and the others, all use really stupidly complicated examples. Note, I'm not saying I'm a genius. But I'd bet, at least one of the other mathematicians I've referred to regarding Bayesian statistics, might be. BUT @Kangal, like Rev. Bayes (who I believe to be a believer) and Pascal (who I'd wager was also a believer), I believe you and your comment to be genius. If there wasn't a virus going around I'd kiss you. 😘
I've watched this video several times and I can't believe I've never noticed this comment. And when I did, I burst out laughing and nearly knocked over my cup of coffee onto a mum with a toddler on the next table (I'm in a cafe)? What's the statistical probability of that (i.e. there being a mum and the toddler while reading a comment about a mum and a [metaphorical? toddler)?
Schrödinger, Descartes and Bayes walk into a bar. The bartender asks: "Would you gentlemen like a drink?" Schrödinger says: "Yes and no." Descartes says: "I don't think so." Bayes says: "I'm not certain."
"And thanks to the owners of the house! Turns out, one of them's a mathematician..." This is the least surprising statement I think I've ever heard in my life.
I've never seen Bayes' fundamental insight so amusingly explained. Nor so clearly. Nicely done! (Also: congratulations to Dr Fry on the forthcoming addition to your family!)
Have you read the paper they are referring to? This experiment is discussed in Section II. It's one of those things you have to draw on paper or do in real life to get a proper feel for what Bayes' saying. What I found really interesting is that Rice's motivation for submitting this paper was for it to be used as proof of Intelligent Design. 😮
@@warrenyazzie9975 Actually I'd say that Bayesian view excludes gods alltogether. There's no confidentialy, that you can measure, about exitence of a physical god. There is good examples of a methaphorical gods exitance though. Look for example all churches and temples and what not and ask if the builders had a god in their mind when they did those. But a mtphorical god is like justice: it exists only in deeds and thoughts of men.
@@user-ky6vw5up9m I think because you can be either random or not. It's hard to be "more" random. You could increase the sample size of course or the step size, but random is random, it doesn't become more random.
@@AtticAurel I was surprised to learn that “random” isn’t well defined mathematically. But probability is, and comparing probability distributions is well researched, to say the least.
It is official YOU DO NEED TO DO MORE WITH BOTH OFF YOU. The interaction (and faces drown) between you guys is hilarius to watch, and we are learning at the same time. Great video. Greetings from Belgium.
Literally bought Hello World an hour before this video was released. Now I have 2 great maths books to get absorbed into. Maths books are like buses, you wait years for a good one and then 2 turn up almost at once!
Do you think Hannah's book is good for actual computer programmers? I would read it if it is more about the ethics and psychology relating to algorithms, but if half of it is trying to explain algorithms in laymen's terms I would not enjoy it.
@@karlkastor It's definitely more about the ethics of the algorithms used in our modern world, and she rarely explains what the algorithms are and how they work (only enough to potentially peak interest so you go off and Google it). It's not a book to read if you want answers and definitions, it's a book that presents questions you didn't even think about asking in the first place. It's terrifying and hilarious, but mostly terrifying; I highly recommend it :).
When I was in high school statistics we were talking about Bayesian probability, and one of my classmates asked "Isn't that a black asian?", and my teacher said "no I believe that's blasian" and I will always remember that interaction.
I know it's been a long time since this came out, but I have to say, the best part of this video is watching two good friends have fun with each other!
A video with two hosts where one is sitting down for most of the video while the heavily pregnant woman has to stand the whole time... Fine work, Matt! In all seriousness, this was a lovely video and the playfulness between you two is infectious.
Apropos of nothing, please thank the owners of that house for letting us glimpse inside. It's beautiful! Also, congrats to Dr Fry. I genuinely didn't notice until I read these comments.
Excellent, excellent, excellent video. I studied Computer Science at uni, with a little bit of AI. Bayesian statistics was important, but I never knew where Bayes was coming from. My eyes are well and truly opened. And of course, many congratulations to Dr. Fry.
I would have loved to see Matt throw the second ball directly on the first ball and Hanna tell him that the coordinates of the balls are exactly the same. Great video! :D
Finally! All that stuff I did in a previous job makes sense. I could do the computer stuff, I could do the mathematics stuff; but now I think I actually SEE what it's all about. Great job. I expect no less from this channel, but this one really struck a chord. 😍Hannah 🤩Matt The wrap-up from 11:00 or so should be a compulsory part of the education system.
Just reading the part of “Hello World” where Bayesian Statistics was discussed, and it reminded me of this video. This is probably what prompted me to buy the book in the first place! Took me a while, but I finally got around to reading it. Gotta pick up Humble Pi next!
If Hannah, Matt, and Grant (Sanderson) did a video together that would be amazing. Such lovely humans who share their unbridled joy of mathematics make my day.
I just wrote an exam on Bayesian/Belief networks. It's a very interesting topic. They are very useful. Like Hannah's self-driving car example, robots can never know for sure if their sensory inputs are correct, so you need to account for uncertainty by using Bayesian nets. Matt could do a video on Hidden Markov Models, which are the simplest Belief networks to understand.
I think it is worth watching Matt’s videos more than once sometimes. I thought he was just being silly updating his belief about the house, but I had forgotten that by the end of the video. But watching again (showing it to my niece) I noticed right away what he was doing.
Really appreciate this because that equation is on my wall as I’m revising statistics for my semester 2 first year exam and this has given me some insight
"If you had an infinite number of throws, and you could throw them perfectly randomly..." he would miss the table most of the time. It's like the old joke (well, I'm old and I was told the joke) of a physicist trying to explain the Monte Carlo method in a bar, so they describe it in terms of throwing darts at a dartboard. Come back next week and the bartender says "welcome back, I loved your explanation. Has your aim gotten any better?"
Speaking of "what were you before"; at least in some corners of science, the opposite of Bayesianism (or perhaps more accurately, the paradigm Bayesianism is generally set up as an alternative to) is Frequentism. I remain to be convinced that there is a meaningful difference in practice.
I feel like there is naturally going to be a bias introduced if the thrower is the one guessing where the subsequent ones are in relation to their first throw. This is because, the thrower is aware that their first throw hit the table, and without being able to SEE the table, will try to mimic their initial throw (because you'll otherwise get a lot of instances of "Did not hit the table try again", and they will be trying to help the experiment proceed). To alleviate this, i suggest adding a third person, who will only throw AFTER the first throw, and therefore will not have that initial throw to base their subsequent throws on, but will still receive the same information.
Used these concepts a lot while studying analog telecommunications. A typical question is: given that I am receiving X what's the probability that Y was transmitted.
When she told Parker that he was being very consistent with his throws, he could tone down the modifications to his x1, x2, x3... values, getting a more accurate result in the process. More rigorously, small deviations in y must result in small corrections to x from my understanding.
It's hilarious how this video manages to explain the concept well even though Matt's terrible throws made it a complete fluke that his guess was close. The first ball *was* in the South of the table, but it could have been slightly North of centre and he'd have still guessed it was that far South because every subsequent throw landed right on the Northern edge.
Am I misunderstanding or is this essentially how we play sports or for that matter any physical activity? For instance you are about to attempt to score a goal in football (soccer). You must judge the position of the goal and the keeper as well as his body position and probable future movements. As well, other players may be a factor with the same considerations as the keeper. You must judge the ball. Is it at rest, rolling or bouncing? If bouncing you must decide at what height you will make contact. You also must judge your own self. Off balance or ideal striking position and your own skill. Weather can be a factor too. In other words, in the physical world we are all Beyesians.
Bayesian statistics is basically how long distance sniping works, as the spotter tells the sniper how far off he was on the previous shot in order to increase accuracy
Im no mathematician but im pretty sure that's not the complete equation. I believe that's only for relationship between 2 distinct objects. You need to use the one with the summation in it. And this one im not sure but every time there is summation, there is a derivation/integration version too.
The term I hear used that is more in contrast to Bayesian is "frequentist" just FYI. This is more looking at the things in a more Gaussian or Fischer way.
I wish my math professors were talented and funny as you: I didn't know any better back then, and I might be "bayesed", but I'm now more confident math can be entertaining. ;) Also, congratulations Dr. Fry!
You have a different concept at play here in the throws with precision versus accuracy. Having so many hits on the far side of the table and generally in line with the target throw though balanced slightly west of shows a sense of precision without accuracy.
You were a positivist, mr Parker. Objectivism means that knowledge is objectively true or false. This is opposed to stuff like social constructivism where knowledge is created and given meaning in a social context (eg. language only makes sense in a specific social arena, where everybody agrees what words mean) Also, this is why the world needs the softer fields, so we don't end up with AIs rampaging through the streets devouring everyone in search of the perfect stamp collection (bonus points if you get the reference)
Statistics is a modern branch of mathematics, usually thought of as of the 19th century. It was only widely applied in fields of high impact during the second world war. Probability was the earliest step towards it. Averages, distributions and average of differences seems obvious and important in retrospect! Did understanding uncertainty have theological inhibitors?
For a second I thought Tree(3)! meant Tree(3)×Tree(2)×Tree(1) and was all "Psssh, easy." Then I realized my mistake and had a second think. Factorials grow quickly, but Tree grows much, MUCH faster. I think adding another step of Tree will blow the factorial operation out of the water.
"Everytime you're throwing the ball you're getting more information about... " where the table isn't
Ah, the Parker Throw...
Parkesian statistics
I don't know why they didn't just play Battle Ships... its based on Bayesian reasoning!
"where the table isn't"
It's a Parker Square table…
@@ekinteko I believe you and your comment to be genius!!! 😱
I believe this based on my never having thought of this example nor have I seen anyone else explain bayesian statistics with such a sublimely simple example. I, and the others, all use really stupidly complicated examples.
Note, I'm not saying I'm a genius. But I'd bet, at least one of the other mathematicians I've referred to regarding Bayesian statistics, might be.
BUT @Kangal, like Rev. Bayes (who I believe to be a believer) and Pascal (who I'd wager was also a believer), I believe you and your comment to be genius.
If there wasn't a virus going around I'd kiss you. 😘
- "What's the chance you find me here?"
- "It's the first time I've found you here, so I don't know"
I'll use that joke, with your permission...
It’s all yours!
That's a very frequentist approach to chances. After his conversion to Bayesianism, Matt wouldn't have said that ;)
well, we know that it's more than 0%
Chance is about predicting so in fact we actually know that the chance of them meeting is 100% since it happened, therefore has become a fact.
"It's the first time I come here" would have been funnier. :D
“How good are you at catch” “I understand the theory” 😂😂😂
My history in experimental physics 😂😂
Hannah is getting plenty of practice as to being a mum of a toddler here.
This might be the best comment I've ever seen on TH-cam
Ahahahaha 😂😂😂 this REALLY got me, for some reason!
I've watched this video several times and I can't believe I've never noticed this comment. And when I did, I burst out laughing and nearly knocked over my cup of coffee onto a mum with a toddler on the next table (I'm in a cafe)? What's the statistical probability of that (i.e. there being a mum and the toddler while reading a comment about a mum and a [metaphorical? toddler)?
“Its okay, we can do it again 😃”
Can I just have Hannah and Matt explain everything to me? The sheer enthusiasm for math on display is infectious.
Schrödinger, Descartes and Bayes walk into a bar. The bartender asks: "Would you gentlemen like a drink?"
Schrödinger says: "Yes and no."
Descartes says: "I don't think so."
Bayes says: "I'm not certain."
didn't get the descartes part
"I doubt therefore I think, I think therefore I am" - René Descartes
Bayes: "Ask us again..."
Descartes: "I think not" *disappears*
I'm going to use this one
Matt Parker: Making a pregnant woman repeatedly bend down and pick up his throws, while he is sitting in a comfy chair :P
I didn't notice she was pregnant
Parker chivalry
@@Peter_1986 You mean like being near a staircase?
I didn't even realize she was pregnant before your comment. Now that I looked again it's so obvious...
@@Peter_1986 I don't think they were saying it was dangerous, just a little rude since it's harder to bend down when you're pregnant :)
"Embrace uncertainty, but quantify it" is my new motto.
This has the same energy as "Take it easy, but take it"
"And thanks to the owners of the house! Turns out, one of them's a mathematician..."
This is the least surprising statement I think I've ever heard in my life.
I've never seen Bayes' fundamental insight so amusingly explained. Nor so clearly. Nicely done! (Also: congratulations to Dr Fry on the forthcoming addition to your family!)
I have heard that throwing the ball on the table example before, but actually seeing it being done makes it so much clearer to understand!
Have you read the paper they are referring to? This experiment is discussed in Section II. It's one of those things you have to draw on paper or do in real life to get a proper feel for what Bayes' saying. What I found really interesting is that Rice's motivation for submitting this paper was for it to be used as proof of Intelligent Design. 😮
@@warrenyazzie9975 Actually I'd say that Bayesian view excludes gods alltogether. There's no confidentialy, that you can measure, about exitence of a physical god. There is good examples of a methaphorical gods exitance though. Look for example all churches and temples and what not and ask if the builders had a god in their mind when they did those. But a mtphorical god is like justice: it exists only in deeds and thoughts of men.
Some Parker throws right there
In fairness to him he did give it a go, even though he wasn't entirely successful
Clearly it was a Parker Square Table. 🙂
4:10 Hannah's exasperated declaration of "Try" should cause you lot to call and thank your mothers.
I love your profile picture and username! I wish more people were familiar with Berzerk.
Hanna Try
6:03 One mathematician tells another mathematician to be more random.
The irony here is delightful.
Micah Long why ironic?
@@user-ky6vw5up9m
I think because you can be either random or not. It's hard to be "more" random. You could increase the sample size of course or the step size, but random is random, it doesn't become more random.
@@AtticAurel I was surprised to learn that “random” isn’t well defined mathematically.
But probability is, and comparing probability distributions is well researched, to say the least.
It is official YOU DO NEED TO DO MORE WITH BOTH OFF YOU. The interaction (and faces drown) between you guys is hilarius to watch, and we are learning at the same time. Great video.
Greetings from Belgium.
Always love to see Hannah Fry! Loved this one as well.
Literally bought Hello World an hour before this video was released. Now I have 2 great maths books to get absorbed into. Maths books are like buses, you wait years for a good one and then 2 turn up almost at once!
Do you think Hannah's book is good for actual computer programmers? I would read it if it is more about the ethics and psychology relating to algorithms, but if half of it is trying to explain algorithms in laymen's terms I would not enjoy it.
quote of the century
@@karlkastor It's definitely more about the ethics of the algorithms used in our modern world, and she rarely explains what the algorithms are and how they work (only enough to potentially peak interest so you go off and Google it). It's not a book to read if you want answers and definitions, it's a book that presents questions you didn't even think about asking in the first place. It's terrifying and hilarious, but mostly terrifying; I highly recommend it :).
Anybody else find Hannahs voice soothing? Intelligent and beautiful but the voice gets me everytime.
"It's all about that Bayes" - Thomas Bayes probably.
There's a paper titled "Drum'n'Bayes". They use Bayesian networks for pattern recognition on music.
"Its all about that Bayes - no table" :-)
I think, you guys may be a little bit biased.
Not about the treble
6:00 "you're actually remarkably consistent"
"Consistency is only a virtue if you're not a screw-up" -Grant Sanderson
Can always count on Grant to come up with the goods!
They always have so much fun together, that it spreads and reaches us through.
This is the first i've seen or heard of Hannah Fry. I'm already a fan. She is awesome.
"I was a certaintist...I don't know."
You don't sound like a very good certaintist.
He was certain that he didn't know!
I know that I was a certainist, but now I'm fairly confident that I am a bayesianist.
Bayesian sounds like a colour. I'm not sure what colour.
@@RaglansElectricBaboon Beige-ian.
When I was in high school statistics we were talking about Bayesian probability, and one of my classmates asked "Isn't that a black asian?", and my teacher said "no I believe that's blasian" and I will always remember that interaction.
I know it's been a long time since this came out, but I have to say, the best part of this video is watching two good friends have fun with each other!
Matt and Hannah are a great team, I enjoyed this video very much.
Love your silly mathloving face! That smirk and side eye gets me every time!
Another Parker/Fry video? Ummm, hell yes! Love you two :D
They should write a joint book titled "Tinker, Tailor, Parker, Fry"
@@_zelatrix I'm sure they could find two mathematicians with apt names who can be given credit as co-authors.
I wish Hannah Fry had a TH-cam channel
At least we've got "The Curious Cases of Rutherford and Fry"
@@ViatorRex Oh! Is that this Rutherford?!
I love this! I'm taking a class on Bayesian statistics now, and I would consider myself a Bayesian vs a Frequentist.
congrats for the baby!
And to the father
Yeah, another person to suffer on this world.
@@LordNezghul oof edgy
@@LordNezghul only Jesus can end our suffering
@@joshuawan7004 lol wtf?
A video with two hosts where one is sitting down for most of the video while the heavily pregnant woman has to stand the whole time... Fine work, Matt! In all seriousness, this was a lovely video and the playfulness between you two is infectious.
She's the expert, so she's the one that came up with the experiment.
Apropos of nothing, please thank the owners of that house for letting us glimpse inside. It's beautiful!
Also, congrats to Dr Fry. I genuinely didn't notice until I read these comments.
Excellent, excellent, excellent video.
I studied Computer Science at uni, with a little bit of AI.
Bayesian statistics was important, but I never knew where Bayes was coming from. My eyes are well and truly opened.
And of course, many congratulations to Dr. Fry.
Parker throw (I had to)
I would have loved to see Matt throw the second ball directly on the first ball and Hanna tell him that the coordinates of the balls are exactly the same.
Great video! :D
Always love videos with Hannah, congrats on her upcoming family too
Their duo is killing it! Good match!
Finally! All that stuff I did in a previous job makes sense. I could do the computer stuff, I could do the mathematics stuff; but now I think I actually SEE what it's all about.
Great job. I expect no less from this channel, but this one really struck a chord.
😍Hannah
🤩Matt
The wrap-up from 11:00 or so should be a compulsory part of the education system.
Wionderful and clear explanation. And, a priceless end skit of sharing about two books.
Mr. Parker I absolutely love your videos! And Hannah being in this one makes it even better!
they should make an audiobook together where they simultaneously read their 2 books...
Oh god the cacophony...
Check out "Numberphile"
Someone should write two books so that they synchronize when read together.
PLEASE I would love it
You guys need your own tv show 💜
The humor is on point. Love the videos
Just reading the part of “Hello World” where Bayesian Statistics was discussed, and it reminded me of this video.
This is probably what prompted me to buy the book in the first place!
Took me a while, but I finally got around to reading it.
Gotta pick up Humble Pi next!
You guys are great, Matt. Love the dynamic between the duo!
If Hannah, Matt, and Grant (Sanderson) did a video together that would be amazing. Such lovely humans who share their unbridled joy of mathematics make my day.
Love the way, Hannah explains things
I just wrote an exam on Bayesian/Belief networks. It's a very interesting topic. They are very useful. Like Hannah's self-driving car example, robots can never know for sure if their sensory inputs are correct, so you need to account for uncertainty by using Bayesian nets. Matt could do a video on Hidden Markov Models, which are the simplest Belief networks to understand.
"As I throw more balls my confidence in my throwing skills lowers asymptotically towards zero."
Both of you keep math fascinating, and you're both loved for it.
I think it is worth watching Matt’s videos more than once sometimes. I thought he was just being silly updating his belief about the house, but I had forgotten that by the end of the video. But watching again (showing it to my niece) I noticed right away what he was doing.
Really appreciate this because that equation is on my wall as I’m revising statistics for my semester 2 first year exam and this has given me some insight
"If you had an infinite number of throws, and you could throw them perfectly randomly..." he would miss the table most of the time.
It's like the old joke (well, I'm old and I was told the joke) of a physicist trying to explain the Monte Carlo method in a bar, so they describe it in terms of throwing darts at a dartboard. Come back next week and the bartender says "welcome back, I loved your explanation. Has your aim gotten any better?"
This is the content I subscribed for and I'm definately not disappointed
these conversation is like two mathematicians flirting each other.
Statistics indicate Hannah Fry will always be a great guest.
Two of my three favourite Mathologers having fun.
You guys look like you're having so much fun!
I love it~
Hannah and Matt are a great combo for making math fun!
Excellent video. Thank you Dr Fry and Matt!
probably my favourite mathematician and my favourite school of inference in one video.. Matt Parker, you've done it again
Bayes rocks! And probably rules too. Thanks for such a fun rundown on the basic approach
Congratulations, Hannah!
Hannah and Matt in one video? Great start into the weekend!
You 2 are absolutely delightful.
Speaking of "what were you before"; at least in some corners of science, the opposite of Bayesianism (or perhaps more accurately, the paradigm Bayesianism is generally set up as an alternative to) is Frequentism.
I remain to be convinced that there is a meaningful difference in practice.
Try this, it convinced me: jakevdp.github.io/blog/2014/06/06/frequentism-and-bayesianism-2-when-results-differ/
These 2 fabulous ppl would be on my fantasy dinner table guest list so much fun thank you
I feel like there is naturally going to be a bias introduced if the thrower is the one guessing where the subsequent ones are in relation to their first throw. This is because, the thrower is aware that their first throw hit the table, and without being able to SEE the table, will try to mimic their initial throw (because you'll otherwise get a lot of instances of "Did not hit the table try again", and they will be trying to help the experiment proceed).
To alleviate this, i suggest adding a third person, who will only throw AFTER the first throw, and therefore will not have that initial throw to base their subsequent throws on, but will still receive the same information.
Used these concepts a lot while studying analog telecommunications. A typical question is: given that I am receiving X what's the probability that Y was transmitted.
The two of you are so entertaining. I really love it. 😁
You should use this uncertainty to save your Parker Square.
This is one of my favorite things on youtube
A fun, interesting, educational duo - the king and queen of mathematical edutainment
When she told Parker that he was being very consistent with his throws, he could tone down the modifications to his x1, x2, x3... values, getting a more accurate result in the process.
More rigorously, small deviations in y must result in small corrections to x from my understanding.
This is the best video I have seen this month.
I heard "try more random" and I couldn't help laughing.
This was great! Thanks for the maths and the laughs.
Many thanks to the stars of this - Hannah, Matt, the briefly appearing camera operator, and of course Hannah Jr ...
Hannah Fry is a treasure
It's hilarious how this video manages to explain the concept well even though Matt's terrible throws made it a complete fluke that his guess was close. The first ball *was* in the South of the table, but it could have been slightly North of centre and he'd have still guessed it was that far South because every subsequent throw landed right on the Northern edge.
OMG two of my favorite people!
8:26 Matt's face when he imagines having an infinite number of throws
Am I misunderstanding or is this essentially how we play sports or for that matter any physical activity? For instance you are about to attempt to score a goal in football (soccer). You must judge the position of the goal and the keeper as well as his body position and probable future movements. As well, other players may be a factor with the same considerations as the keeper. You must judge the ball. Is it at rest, rolling or bouncing? If bouncing you must decide at what height you will make contact. You also must judge your own self. Off balance or ideal striking position and your own skill. Weather can be a factor too. In other words, in the physical world we are all Beyesians.
Loved the plugs at the end, very nice indeed
Bayesian statistics is basically how long distance sniping works, as the spotter tells the sniper how far off he was on the previous shot in order to increase accuracy
Great science communicators! Science needs more Matts and Hannahs!👩🏫👨🏫
Im no mathematician but im pretty sure that's not the complete equation. I believe that's only for relationship between 2 distinct objects. You need to use the one with the summation in it. And this one im not sure but every time there is summation, there is a derivation/integration version too.
The term I hear used that is more in contrast to Bayesian is "frequentist" just FYI. This is more looking at the things in a more Gaussian or Fischer way.
and now on Kolmogorov statistics!
I wish my math professors were talented and funny as you: I didn't know any better back then, and I might be "bayesed", but I'm now more confident math can be entertaining. ;)
Also, congratulations Dr. Fry!
You have a different concept at play here in the throws with precision versus accuracy. Having so many hits on the far side of the table and generally in line with the target throw though balanced slightly west of shows a sense of precision without accuracy.
You were a positivist, mr Parker. Objectivism means that knowledge is objectively true or false. This is opposed to stuff like social constructivism where knowledge is created and given meaning in a social context (eg. language only makes sense in a specific social arena, where everybody agrees what words mean)
Also, this is why the world needs the softer fields, so we don't end up with AIs rampaging through the streets devouring everyone in search of the perfect stamp collection (bonus points if you get the reference)
Statistics is a modern branch of mathematics, usually thought of as of the 19th century. It was only widely applied in fields of high impact during the second world war. Probability was the earliest step towards it. Averages, distributions and average of differences seems obvious and important in retrospect! Did understanding uncertainty have theological inhibitors?
Matt, if you had to guess, what would you say is the bigger number:
A: Tree(3)!
B: Tree(4)
tree 4
For a second I thought Tree(3)! meant Tree(3)×Tree(2)×Tree(1) and was all "Psssh, easy." Then I realized my mistake and had a second think.
Factorials grow quickly, but Tree grows much, MUCH faster. I think adding another step of Tree will blow the factorial operation out of the water.
Factorial is actually pretty weak once you get up to numbers like Tree(n>2). It's like lending a dollar to a trillionaire.
@@ryanoftinellb *Tree(llionaire)
ignore my ignorance... *ashamed in silence*
Nice explanations. It will be very useful to me! Tks.
Bayes and Fry in a single episode. How much more can a poor boy stand and live. Brilliant.
That outro was the cutest thing!
„february”
ah, love all those fresh videos
All the awesome people are writing books these days!
Am I the only one getting strong Beverly Crusher vibes from Hannah in that jacket?
Well, I am now
Hush Wesley.
I didn’t the first time but just watched this again, and that’s the first thing I thought when I saw Hannah!