Bayes Billiards with Tom Crawford

[@wrh41]

lol! No one could take your place as the most beloved presenter of Numberphile, James!

[@mkaiww]

I thought Matt was the most beloved presenter of Numberphile :)

[@Henrix1998]

@@mkaiww Hannah Fry, Tadashi Tokieda (origamis), Cliff Still (Klein bottle) and the list goes on, everyone is just as loved

[@bruinbear7666]

Henrix98 Except the -1/12 guys.

[@wrh41]

@@bruinbear7666 lol yes - suuuuuuure, it sums to -1/12 ... suuuuuuure. they'd make good used car salesmen!

[@Snaperkid]

wrh41 I does sum to -1/12, kinda. It all has to do with analytical continuation which asks: “This is a divergent sum. If this sum had a value what would that be the most logical value of the sum?” Sometimes it turns out that infinity is the most logical value for the sum like in the case of the harmonic series. However other times you can find a value that makes logical sense. I would recommend watching Mathologer’s or 3B1B’s video on it for more details.

[@kasajizo8963]

*I can't believe after like 10 years he actually uploaded*

[@HasekuraIsuna]

More like 10 months, but I get the feel bro

[@Saboo8444]

Well, it's confirmed. Thomas is a terrible random number generator

[@singingbanana]

I think this is the main takeaway.

[@fsmvda]

Yeah, he was looking at it when he did it. Humans like to make things evenly spaced. Like how people don't put enough long repeat runs if they try to write a list of random coin flips.

[@mrmonster3434]

Two mathematicians walk into a pub. They spend a while playing with the pool table, then complain there is a flaw in the Bayes.

[@Mikeastro]

I just love how they're clearly just having so much fun filming this

[@singingbanana]

It was fun.

[@TomRocksMaths]

@@singingbanana Agreed.

[@madhuragrawal5685]

Why don't you pull a matt parker and run a simulation in the most accessible programming language, excel?

[@ChickenGeorgeClooney]

Or just call it a Parker Prediction and call it a day.

[@JayTheYggdrasil]

I don't know what your talking about, most accessible programming language is SQL

[@ukko1998]

@JayTheYggdrasil the most accessible programming language is VBA

[@ozapenguin]

@@JayTheYggdrasil it's a joke

[@JivanPal]

@@ozapenguin, *_whooo ooo ooo oo oo oo o oo ooo o o ooo o ooo o o o oo oo oo o o oo oo o o o o o oo o o oo sshshsshhshhhhh_*

[@emkin6128]

00:12 "I know you're trying to take my place as the most beloved presenter of Numberphile" *Matt Parker has joined the chat*

[@xevira]

"Why has Boss music started playing?"

[@rawovunlapin8201]

I present to the jury: Cliff Stoll

[@leonhardeuler9839]

Rawov Un Lupin “I see you are a man of culture, as well”

[@GuanoLad]

Joining the long list that includes Hannah Fry, Holly Krieger, Ben Sparks, Alex Bellos, and Cliff Stoll.

[@shubhamg9495]

Cliff Stoll: **Ahem**

[@alwysrite]

The probability of someone identifying Tom as a Math professor is close to zero ! Never judge a book by it's cover : )

[@TimothyReeves]

More amazing: the probability could actually be zero, and he could still be a math professor!

[@golagaz]

One of my friend is a professor of applied maths too, he looks like a football hooligan :) lol

[@nekogod]

His tattoos give the game away a little bit, he has several mathematical tattoos.

[@dragoncurveenthusiast]

Thank you for mentioning Tom's channel! I wasn't aware!

[@Raykkie]

The throws weren't independent ! We need a rematch !

[@OceanBagel]

True academics, not redoing the experiment to get the result they expect. Don't do publication bias, kids!

[@singingbanana]

Thank you!

[@RUBBER_BULLET]

Or they didn't have the time or inclination to repeat it. Don't project your own biases, kids.

[@thewiirocks]

Or they could have repeated the experiment multiple times on camera to show what the effects would be. Their sample size is too small.

[@pegy6384]

Your props are getting fancier, Dr. Grime! Very much enjoyed the video, and glad to see you back--it's been a while.

[@singingbanana]

Thank you!

[@sk8rdman]

"Hi everyone! We're going to do some statistics today, which is kind of exciting." Only in this mathy corner of KZbin is a video going to start with that sentence, and have the viewers get genuinely excited.

[@planetx1595]

So maths bored you to death at school? I must admit, the teachers I had didn't make the subject anymore fun.

[@sk8rdman]

@@planetx1595 I think you misunderstand me. I'm no different from the other viewers who got genuinely excited when he mentioned statistics. I just acknowledge that the prospect of "exciting statistics" is not something most people can sympathize with. I'm glad to be among the audience of this channel where we can share our love of mathematics with genuine enthusiasm.

[@WatchingTokyo]

A guy goes into a bar and says "Hey everyone! It's me!". Everyone turns around just to realize that it wasn't him.

[@GHollandESQ]

American: What rhymes with orange? Brit: No it doesn't.

[@mdhedaetulislam9009]

I am sorry to tell you guys that, there is a flaw you did not consider. that is, the red ones collided among themselves and these internal collisions make the individual red balls' position interdependent....so the positions are not an independent event....they're very much dependent. Suggestion: please conduct the experiment once aagain. This time you should throw the red balls one by one and mark the position by a chalk on the board.(hopefully, the marks by chalk will not affect the experiment)

[@singingbanana]

You're absolutely right.

[@illiros]

also the rails cause the balls to lose a fair bit of energy, leaving them more likely to be closer to a rail than in the center of the table

[@NoriMori1992]

I hope it's okay that mine's more of an inductive reasoning joke than a math joke: Three logicians walk into a bar. The bartender asks, "Do you all want a beer?" "I don't know." "I don't know." "Yes."

[@singingbanana]

I like this one a lot.

[@Opaqu.e]

Oh! My 4th and 9th favourite numberphile presenters in the same video!

[@Wyldina]

Who's better than James, besides Dr. Holly?! :O

[@stormsurge1]

@@Wyldina Maybe Tadashi Tokeida and Cliff Stoll?

[@hirakmondal6174]

@@stormsurge1 U cant rank Cliff Stoll, he belongs to a perfect universe where there is no ranking.

[@JohnLeePettimoreIII]

Burn Level: ALPHA

[@JohnLeePettimoreIII]

@@hirakmondal6174 The good Dr Stoll defies any rankings, agreed. Been my hero for many years.

[@pronounjow]

Dude, I was thinking about you yesterday, and here you are with a new video the very next day! Also, this is the first Bayesian video I've watched! Hurray! :D

[@theniborris]

James, I have 47 math jokes. They're all prime examples.

[@singingbanana]

Thanks!

[@EricPetersen2922]

Welcome back! Great to see you! Missed you!

[@imnotselma3305]

A red-shirted Englishman named Tom

[@deekshas3936]

Ha!

[@Noam_.Menashe]

He is like a sponge though, scott clean.

[@Ultrasonix3]

Alt. universe tom scott

[@dragoncurveenthusiast]

I appreciate the formulas in the video! Glad you decided to make your own version!

[@loolka2029]

I love dragon curves too :)

[@maxhaibara8828]

Isn't he the Fluid Dynamics guy?

[@TomRocksMaths]

Sure am!

[@rickseiden1]

Dr. James, you have nothing to worry about. No matter how popular Dr. Tom is, he'll never be the Jack of Hearts!

[@tscoffey1]

You said that the 14 balls are independent events. But at least two collisions occurred. Isn’t a collision of two balls by definition a dependent event?

[@jivingwatermelon4884]

Presumably you can hear the cue ball bouncing off the sides, and therefore slowing down as it goes up and down the table. How easy would it be to work out the final position of the cue ball based on this?

[@tylern6060]

Mr. Doppler was wondering why the baseball was blue... Then it hit him.

[@singingbanana]

Haha. Good one.

[@BorgusBingo]

this was basically the first 2 hours of lectures of my 3rd year stats course at uni lol, nice one.

[@shruggzdastr8-facedclown]

It's the Navier-Stokes guy!!

[@shruggzdastr8-facedclown]

I love how he correctly corrected James' usage of "true" with "correct"!

[@alexf2008]

Great video as per usual. Thanks for signing my copy of On The Road the other day. Hope to see more content soon!

[@kyleglaeser3361]

An infinite number of mathematicians walk into a pub. The first orders a beer and the second orders two beers. After the third mathematician orders three beers, the bartender tells the lot of them, "you owe me one-twelfth of a beer."

[@singingbanana]

Oof! I love it.

[@t71024]

@@singingbanana Watch your signs! Actually, the bartender owes to the mathematicians.

[@NoriMori1992]

Not really in the same field, but this is my current favourite bar joke: Three logicians walk into a bar. The bartender asks, "Do you all want a beer?" "I don't know." "I don't know." "Yes."

[@Mazzzer98]

Very nice, best video I recently saw. Thank you guys

[@Fr4nC3zU100]

Hi guys, if i have some questions regarding corelations between Bayes and Benford's law how can i find my answers :)? Or where? Or will it be possible to address those questions to you guys? Thanks!

[@WilliamRussell314]

I asked the world expert on bonsai trees what he knew. He said very little.

[@singingbanana]

Thanks!

[@slolilols]

😂 lol

[@MrYashraj]

A Person With Exceptional Skill In A Particular Area❤❤❤.

[@karannchew2534]

10:05 Why getting the average position (sum of probability of position x position) instead of the position of highest probability? "Average position" does not seem to make sense, what do i miss?

A German, an American and a leprechaun walk into a bar. The leprechaun says, "Wait! I'm in the wrong joke!"

[@singingbanana]

Thanks!

[@loolka2029]

James, you are subscribed to Vihart. Do you know where she is now? And is she going to upload new videos? Ps you're back it's sooooo cool

[@kqp1998gyy]

Thank you guys. You are awesome

[@OlliWilkman]

For us physical scientists who cringe at the lack of uncertainty estimates: the posterior standard deviation in this case is (k + 1)/(n + 2) * sqrt((n - k + 1) / ((k + 1)*(n+3))), where k is balls on the left and n is total balls (k=8, n=14), so the Bayes solution with standard deviation would be 0.56 ± 0.12, i.e. the answer is pretty likely to be in the interval [0.44, 0.68]. Not a very precise estimate, but 14 points isn't a lot of data. (I also have some doubts about the true randomness of the way the balls were launched.)

[@fowlerj111]

I was seeing way too many decimal places for the measurement as well as the calculations, at least in the absence of confidence intervals 😁

[@guest_informant]

Knock Knock Knock Knock Knock Knock Knock Knock Knock Knock Knock Knock Who's there? Fibonacci

[@singingbanana]

I love this.

[@user-zz8ln3uh5x]

I wonder if the fact that rails/bumpers absorb energy, pool balls tend to stop rolling/accumulate closer to the rails than the center of the table. Seems like they would.

[@jakebyron2581]

Is this in the Queens’ college MCR?

[@singingbanana]

It is.

[@TomRocksMaths]

Nice spot!

[@Tombsar]

You can tell this man knows his stuff from his apostrophe placement. ;)

[@johnkesich8696]

What is the probability that Frequentist(?) statistics beats Baysian statistics?

[@unvergebeneid]

I just ran a simulation, see my comment asking the same question :D

[@johnkesich8696]

Link to Penny Lane's comment with simulation results kzbin.info/www/bejne/qKjXdXWeeZuAp5I&lc=UgzUC4MfbeHpIPQsBSR4AaABAg EDIT: For some reason clicking on this url ignores the &lc parameter. Copy and paste works.

[@NoriMori1992]

@@johnkesich8696 Yeah, I've noticed that issue as well.

[@DancingRain]

You mean Tom form Equations Stripped? :P

[@TomRocksMaths]

That's me!!

[@MathTravels]

This is great! Does anybody know where David Spiegelhater and Brian Cox video can be found?

[@singingbanana]

kzbin.info/www/bejne/Y5abqIKZlsamma8

[@MathTravels]

@@singingbanana Thanks!

[@Tone720]

As is often said in tournaments of cue sports: Let's get the Boys on the Bayes...

[@mrmonster3434]

If you look at the final distribution of the field of red data points, it seems that some configurations contain a lot more information about the white ball's position than others. In fact, given that the probability of the white ball ending between any two adjacent reds is greater the larger the gap between them, the most likely outcome is also that which allows for the least precision. I may be misunderstanding the experiment, but if it is known whether a ball is to the left or right of the white, surely the best guess for the white's position is the mean of the nearest left and right reds?

[@peppers1587]

Welcome back Mr. Banana and friends.

[@burke615]

It would be interesting to see this re-done, but with truly independent red balls, and have the Bayesian calculation redone after each new ball is thrown. That way it would demonstrate how the Bayesian probability feeds back into the calculation.

[@shadowsfromolliesgraveyard6577]

Wouldn't it be more bayesian to do one red ball at a time, updating the probability for each?

[@singingbanana]

It would. It took ages.

[@BrunsterCoelho]

Hey @ singingbanana, huge fan, just a quick question: The fact that you used Bayes Theorem during the 2nd part, doesn't make the experiment "baysian" right? Ie, both frequentist and baysians accept that theorem as a truth; What makes it a baysian approach is the fact that you included a prior in your calculations and behind the scenes used MAP instead of MLE (right?) PS: Why should you worry about the math teacher holding graph paper? She’s definitely plotting something.

[@singingbanana]

Correct. (And thanks)

[@aladinnotrebienaime1675]

evry time there are Something good to learn on this channel

[@Maninawig]

Bayes Law is something I get the theory of, don't understand the formula, but am still facinated by.

[@Icenri]

James closed the discussion with a checkmate!

[@bretterry8356]

Frequentist may have been slightly closer this time, but you only have a sample size of 1. If you repeated this test 1000 times, would you expect frequentist or Bayesian estimates to be closer on average?

[@alkhwarizmiagainstthemachi4201]

Singing Banana... best KZbinr ever!

[@fernandopaul1]

isn't there a systematic problem in the way thomas trow the balls? would it interfere with the prior P(p)?

[@TheTwick]

Very good, guys. Mrs Bayes would be proud. ;-)

[@alexandersanchez9138]

3:52 I think since the cue ball had landed so much closer to where the group of 6 ended up than to the group of 8's position, the evidence you used to update the prior wasn't good. Maybe a more robust measurement to use would have been the weighted (by the relative rank of each red ball) average of the distances between the cue ball and each red ball. Of course, this calculation would be trickier to do for the purposes of illustration. Am I off-base, here? What do people think?

[@U014B]

Bayes' Law kinda reminds me of the definition of a derivative. Do you think there's some correlation between the two?

[@darrellplank4518]

It seems to me that one of the biggest sources of errors was that you have "sinks" in the form of pockets where the balls go in and don't escape from and the two far likeliest sinks are on the far side of the table where two balls did indeed go into. I feel like those cases should have been tossed as outliers and the answer would have been 8 and 4 instead of 8 and 6 in which case Bayes would have given a more reasonable 0.643. The rest could be explained away as people below have stated by the non-uniform probability distribution of where balls would end up when rolled from a single side, energy dissipation, etc.. Nonetheless, good to see somebody accepting the seeming whims of the probability Gods and not tossing their results just because they don't match original expectations. Fun video! Viva la Bayes! I love what both of you are doing every time I see you doing ANYTHING!

[@petergorelov6852]

Is it possible to do "beta-distribution generator" with billiard table? I mean: we make first experiment with balls on the billiard table -we make calculations - we have one value from beta-distribution. We make second experiment with billiard balls -we have other value from beta-distribution. If your answer is "yes" -please describe whole procedure.

[@drdca8263]

So, is taking the expected value of the posterior distribution what we want? What does the median give? What if we looked at like, an interval for which we would be 90% sure that it would be in? (I suppose the smallest interval for which we would be 90% confident?)

[@andytroo]

whats the probability that the Bayesian estimation is worse than the frequentest one?

[@sabriath]

The speed of the ball due to friction creates a asymptotic projection across the statistic "landing zone".....this heavily changes the idea that "each position is likely." You will have to perform this numerous times in order to work out first the probability of each zone, THEN you can use the formulas correctly. Just saying.

[@pegy6384]

A Bayesian statistician falls down from the top of the Empire States Building. On his way down, he sees a friend watching him from the third window floor. The friends asks how things are going. The Bayesian answers: "Based on my priors, I will be just fine".

[@singingbanana]

This is a good joke.

[@RobertCardwell]

What does a bass player say when he gets to his gig? Would you like chips with your order?

[@singingbanana]

Oof.

[@shruggzdastr8-facedclown]

I saw a video about a year ago on Bayesian statistics/Bayes' Law on the Veritassium channel!!

[@JohnGormleyJG]

Nice to see a classic singingbanana video.

[@KX36]

I don't get it. What do the disembodied legs in the background represent?

[@TomRocksMaths]

You weren't supposed to spot those...

[@user-vn7ce5ig1z]

• 0:12 - I'm not sure what the rankings are or what position Tom is currently in, but I do know there's some stiff competition, and that he'll have to go through Matt before he can take on our Grimy-boy. • 0:48 - James has too much integrity, but also too much paranoia. Many KZbinrs have released similar videos close to each other. Moreover, viewers here aren't JakePaul-weens or Jaystation-twerps, we're smart enough to understand that. • 2:16 - Except that not all positions are equally likely. :-| • 4:08 - Um… am I missing something or should 14-8 have been an easy calculation? 🤨 Was he account for balls on the line? 🤔 • 4:27 - I too have 8 balls on the left and 6 balls on the right… and yet, I still can't have kids. :-\ • 10:20 - When someone asks if you're a god, you say yes! If someone thinks you did the math in your head, you don't deny it. 🤦 • 12:38 - Statistics may be a lie, but probabilities is a filthy scumbag and not true mathematics. 😒 • 13:37 - Baysian statistics is indefensible because I _still_ get spam after 20 years. ¬_¬

[@BusterAtlass]

I wonder if all the red balls being rolled from the left towards the right would skew the randomness of the red distribution and cause an unnaturally high amount to end up to the right, thus pushing the likely position probability down to the left. My thinking is this... a ball rolled from the right to the left is more likely to hit a ball already on the table and push it further right than it is to miss all the balls, bounce off the cushion and hit a ball as its travelling right to left and push it left. This is more of a gut instinct than anything definite and I would be interested to hear yours, and viewers, thoughts on this.

[@GuanoLad]

One red ball was on the line. What if we change to either L=9 R=5, or eliminate that ambiguous ball and make it L=8 R=5?

[@st0rmforce]

Do you think that the fact that you threw all the red balls out at once made a difference? If you'd thrown them one at a time, would they have landed in more naturally random places? You're assuming that all positions are equally likely, but could there be clumping due to collisions?

[@mattmerr]

How do statisticians fix dents on their cars? By adding more until it looks normal.

[@singingbanana]

Ha! I like it.

[@avikdas4055]

We need more of James in numberphile. The guys who boosted the channel are missing nowadays like James Grime, Simon Pampena, Simon Singh, Hannah Fry etc.

[@stumbling]

Surely the balls cluster around the cushions?

[@prdoyle]

What if the initial distribution wasn't uniform?

[@BillySugger1965]

The distribution of red balls was nowhere near random; most bounced off the RH cushion only once which skews the distribution to the right. The cue ball rebounded several times, and its final position was therefore more representative of a continuous even distribution. I’d be really interested to see a follow up to this with better control of the distribution of ball positions, and see whether you get a better demonstration of Baysian statistical methods. This topic is so important to the statistical analysis of experimental results, surely it must be worth another go, with better controls?

[@BillySugger1965]

A better method would be to shoot each of the red balls with the cue, just as you did with the cue ball, bouncing at least three times to be valid, measuring its position and removing it before shooting the next ball (some fancy editing with a fixed camera can show the distribution of all), and firing alternate balls from opposite ends of the table.

[@lilyrooney]

The thumbnail to this video looked like a "blues clues" style thing to me at first glance... I think i must still be getting over the flu because now that I've clicked the video I can see the background isn't greenscreened in at all hahaha

[@EvilDoener]

What did the mermaid wear to math class? An algaebra. Edit: I almost never leave a comment, but if I'm posting a joke for you, I might as well use the opportunity to tell you, that I've always really enjoyed your videos. I am a math nerd since pretty much forever and love solving puzzles or even just solving math exercises, but your enthusiasm for mathematics is something I always looked up to (even as someone who already loves math). Keep up the good work!

[@singingbanana]

Thanks! And thanks for the top joke.

[@juuke84]

Nice to see you! Its being a while :D

[@superjugy]

James we need you back at numberphile, your videos are the best!

[@pseudomonad]

Baize Theorem?

[@MrV1604]

Ha! quite underrated

[@unvergebeneid]

Now I wanna know what the probability of the frequentist approach being closer to the true value compared to the Bayesian one is. *Edit:* Answer in the comments.

[@unvergebeneid]

@@palmomki not yet. I thought someone might have an exact answer here.

[@unvergebeneid]

@@palmomki I did now. Here's the output of my script: simulated 14 balls 10000000 times the Bayesian approach won over the Frequentist one 50.44934% of the times 6.65762% of the simulations were ties, so 42.89304% were Frequentist wins the mean deviation from the real value for the Bayesian approach was: 0.0800043179873167 the mean deviation from the real value for the Frequentist approach was: 0.08422584547321381

[@unvergebeneid]

The code is not really worth a github but here it is for others to play around with: pastebin.com/0xCc6mW1

[@unvergebeneid]

And to see how these values change with the number of "data" balls used, I extended the script a bit: pastebin.com/yyCjtnEu and here are the results: imgur.com/a/MkdQVIi The see-saw pattern stems from the fact that ties happen iff l=r, which is obviously impossible for an odd number of balls. Since tie_ratio + bayes_ratio + freq_ratio = 1, the tie ratio affects both win ratios as well.

[@unvergebeneid]

Yes, the Bayesian approach is clearly superior but not by much. For this video to have a non-desired outcome (for a given value of "desired") was pretty much a toss-up and the whole affair converges towards toss-up with the number of balls increasing. Interestingly, because of the odd-even effect, one _fewer_ balls would've slightly _increased_ the chances of a Bayesian win (from ~50% to ~54%). So, odd choice there, pun not intended.

[@MrMcCoyD4]

If you did this a million times which strategy would get you close more of the time? Is the point of the video supposed to be that Bayesian is better?

[@singingbanana]

I believe the idea is that Bayesian is better with limited data.

[@AndrewTaylorPhD]

Ah, yes, Baize Theorem.

[@sungodmoth]

Great video as usual! I enjoyed your talk at the university of exeter in December :p

[@singingbanana]

Hi again! I enjoyed it too.

[@sumankumarshah6176]

that's the coolest looking maths professor ever

[@CastorQuinn]

Agreed. Tom looks pretty cool too though.

[@mestiarcanus]

The better defence would be that the red ball locations weren't independent because the balls were allowed to interact.

[@singingbanana]

I think you're right.

[@seventhtenth]

5:47 "factorial thing" = binomial coefficient

[@chih_le]

it was correct to consider that not every position is equally likely, but the distribution should be shifted based on the shortest and longest distances travelled by the balls from where they were released

[@35571113]

The discussion at the end was such an excuse to find a standard deviation (for the Bayesian approach) and a confidence interval (for the frequentist)...

[@singingbanana]

Exactly.

[@siddhantkumar6340]

you are my favorite numberphiler

[@GoranNewsum]

Perhaps the red balls were knocking into each other too much, if you had done the experiment with one ball at a time, and marked them individually you might have got a better answer?

[@singingbanana]

This might be true.

[@Varksterable]

Seems the balls in pockets still counted as 'right'. Surely that's a pretty big deal, as it's more likely potted balls are to the right as some may not even get all the way back to the original cushion? Pretty hard to get truly random positions for the red balls when only one or two bounces are involved, but the bias always has to be to the right if pocketed balls are counted as righties. So it's not a fair table. (Ignoring pocketed balls gives 0.64, which is closer.)

[@rmsgrey]

Didn't he rethrow the potted balls?

[@TomRocksMaths]

@@rmsgrey I did indeed

[@nugboy420]

You two Along with I can’t remember I’ll probably remember his name and come back on my favorites on Numberphile I was voice texting but back to say Neil Sloan. Tom, James, and Neil. The dream team of Numberphile.

[@powerdriller4124]

The balls were thrown from just one side only, with force enough to make most of them to bounce back to the throwing side. It caused both Bayesian and Frequentist predictions to be way off. A prior was needed. Also, 14 is big enough sample size to get a priorless Bayes to agree with his Frequentist rival, 0.56 is about 0.57.