Randomness

Welcome to the Infinite Monkey Cage, a science show that offers no guarantees due to the uncertainty of the world and the universe beyond it, both statistically and in a quantum mechanical sense.

So, this all could be conjecture, but good, solid, empirical conjecture, nevertheless.

I'm Robin Ince.

And I'm Brian Cox.

This week we're going to be looking at randomness, probability, and chance.

So, in the absolutely literal spirit of one possible meaning of the potentially infinite set of meanings, both real and imagined, of Infinite Monkey Cage, we took a tombola wheel with all the letters of the alphabet on it, spun it 21 times, noted down each letter, and created a title.

It was the Infinite Monkey Cage.

What are the chances of that?

1 over 26 to the power of 21.

Yeah, and I should actually be honest and say that Infinite Monkey Cage wasn't the first set of letters that came up.

The first one was actually you and yours, but that seems ridiculous.

To help us discuss randomness, probability, and chance, we kept with the theme and pulled names out of a hat to decide who the guests would be.

And then we decided it wouldn't really work with our guests, Lindsay Lohan and the Lord Privy Seal.

Anyway, we couldn't get the Lord Privy Seal's number, and also it turned out it's the hat used by I'm a Celebrity, Get Me Out of Here.

So, Lord Privy Seal is gonna be pretty exciting in that locust round.

Well, our first guess is a one in 23 chance of having a number one single, a one in 47 chance of refusing to go on stage if he has a Varuca or a corn, but a one in 3,975 chance of saying, The thing about my crystal is that without it, my chakra just goes haywire.

Now, as a

composer of the new musical of Roald Matilda, and about to embark on an arena orchestral tour, rationalist musician and comedian Tim Minchin.

Our other guest is one of the few people to have written a book on mathematics and ghost-written the autobiography of a world-famous footballer.

Though Bertrand Russell did co-write Principia Mathematica and attempted to ghost-write an autobiography of Nobby Stiles.

It's the author of Pele and Alex's Adventures in Numberland, Alex Belos.

Alex, I suppose at the simplest level, probability is the study of chance.

And I suppose the natural reaction to that is to think, well, chance, completely random events, how can you possibly study it?

It must be entirely random.

So how does a mathematician begin to study chance?

Well, that's the thing.

We can't predict the future for one event.

So if there's something that's got a 50-50 chance of happening, tossing a coin, we don't know what's going to happen.

But if we were to take a million events, toss a coin a million times, we can be pretty pretty sure that round about 50% of the time it'll be heads, 50% of the time it'll be tails.

So once we have a mathematical language, we can understand

where probability is going to go in the long term.

One of the problems with once it actually gets to numbers, generally, there is something that bamboozles the human brain right from the start.

What is it?

What do you think it is about the human brain that makes for some people the understanding, the comprehension of the meaning of numbers?

Well, I mean, there are two things.

One is numbers, and one is probability.

I think lots of people, mathematicians, understand numbers.

You know, lots of people here will understand numbers.

But probability is really, really difficult to understand.

And mathematicians, it's full of pitfalls and mistakes.

So, this idea that I might come along and be able to explain probability to you

exactly.

To be true, absolutely.

You know, I was going to say there's a disclaimer.

C.S.

Pierce, a very famous American mathematician, said, you know, in no other branch of mathematics is it so easy for experts to blunder as in probability theory.

So, you know, if I make lots of mistakes, that means I'm a good mathematician.

Tim,

your songs, I suppose, many of them are based on irrational beliefs, irrational thought.

So, Grashu, do you have any songs about probability to start with?

Well, no, I don't have a probability song.

Why?

There's a reasonable chance I'll write one in the future.

You know, I I've got a song called If I Didn't Have You, which is uh about love and the notion of fate and soulmates and stuff.

So that's got lyrics in it, like

Your Love is one in a million, you couldn't buy it at any price, But of the 9.99900,000 other possible loves, statistically some of them would be equally nice.

And it also says

I think you're special but you fall within a bell curve.

So you know

there's yeah I quite often I find myself saying what are the odds in my shows to make the point that they're reasonable.

Generally the answer to that question is like you're one over 27 to the power of 21 or whatever.

That you can find them eventually.

Have you ever written a song and thought this is a great song but it's actually statistically inaccurate and therefore because that's the thing is you are involved and you write about rationalism, you write about science, you write about, so you actually go, I've got a problem.

I can correctly rhyme this, but this will make it inaccurate.

Or yeah,

there's two things.

One, I do have an obsession with making my songs thorough, which is why they're usually about two minutes longer than is fun.

And the other thing is I try to keep myself just sort of just stupid enough so that I can justify being stupid.

Which isn't to say I need to work very hard to keep myself that stupid.

I just mean I try to make it apparent that I'm not actually claiming to know anything.

Alex, the history of the study of probability is interesting to me.

I know you deal with it in your books.

I mean, it's incredibly recent.

I mean, it's probably the most recent great idea that mathematics has had, which is this idea that we can sort of predict the future using maths.

And it's maybe 500 years old when it was a gambler, in fact, Gieralomo Cardano, an Italian, who really was the first person to think about probability, about games of chance and gambling in a mathematical way.

Because in Rome, for example, people used to gamble all the time, flip coins, and they would think that if Caesar's head came out top, it was Caesar who had decided that you were going to win.

So, probability and randomness was basically sort of superstition.

And superstition died not with Nietzsche or with Darwin, but with Cardano, who said, Actually, we can work out numerically what the probabilities are of flicking a coin.

And he was basically looking at the mathematics of gambling, of gaming.

So he was trying to work out how to design a game such that you could

do something.

Cardano was probably the most interesting person in math history.

He was a doctor, he wrote 131 books, and he also invented lots of interesting maths, including probability.

And he did this because he was an inveterate gambler and he realized that he was losing lots of money.

But there were mathematical ways to actually start winning money.

So people would gamble using dice the whole time.

And he was the first person really to realize that six-sided dice, the chance of throwing a six, is a sixth.

And then you can do the maths like that.

And that seems to us so obvious.

But he was the first person to realise that.

And once he realized that, he started making a lot of money and losing it again.

When you talk about working out probability and when you talk about decisions that you can make and rational decisions, could you, for instance, Tim, live your life by going, hang on a minute, right, I'm just going to work out what is the probability that if I take that action, that will lead to that, and that's the required moment, or does it in turn, does it become a mathematical exercise in living?

I think I do live my life like that.

It's in my nature to try to shed any superstition from any decisions.

I actually consciously work on making sure I've got no superstition left.

The thing I always try to do if a loved one's getting on a plane is say, I hope you have a crash.

Just because I like taking control of what is a very difficult thing.

difficult instinct.

The toughest superstition I've got that I've had to try to rid myself of is the touch wood superstition.

The idea, I go, I've never had a car crash.

Oh, you know, as if your words can change the universe, but it's so embedded in us that we think we're special.

We basically think we're special.

I think it's totally fine to have these little superstitions to make people feel better, to be able to fly easier.

It's just when you lose all your money because you go gambling, it becomes a problem.

And misunderstanding of probability means that people can be conned really easily, and lots of people are conned.

Alex, you tell the story of the way that

our natural sense of coincidence and probability can mislead us.

And you tell the story of the woman who won the lottery in New Jersey twice in four weeks?

Yeah, in four months, I think.

Four months.

So two lottery wins in four months.

Yeah.

And the newspaper...

The newspaper said this was a one in 17 trillion chance of that happening.

And it was a one in trillion chance of...

any person going and buying one ticket on that day and then going and buying the other ticket.

But that's not the way probability works.

If there are thousands or millions of people people actually buying lottery tickets, it turned out mathematicians did the math on it, so to speak.

And the chance of any one person winning two lotteries in America in any four-month period is about 25%.

So it's actually quite

a probable thing to happen over a course of a few years.

It's completely counterintuitive, isn't it?

Which I think is perhaps the origin of superstition and a misunderstanding of many events that happened.

You think there's no chance of that.

You bump into a friend that you've never seen for 10 years walking down the street in London, you think it's a sign.

Coincidences happen a lot more than you think.

And the most famous way of sort of showing that coincidence happen is what's called the birthday paradox, which isn't a birthday paradox at all, which is how many people do you need to have in a room together for it to be more likely than not that two share the same birthday.

We're going to get to that.

Yeah, we are.

Yeah, birthday paradox.

Can we have a little bit later?

Birthday paradox.

Later on in the show.

If you'd sung that for just two seconds longer, we would have had to pay you royalties.

What a pity you missed that.

Birthday paradox, ox ox, ox.

I don't have a problem with that lotto example.

The lay idiots way I think of that is that there's very, very low possibility of thing A happening, but there's loads and loads and loads of things.

Therefore, the probability of any thing happening is really, really good.

In fact, given enough time and enough things, the probability of anything happening is always one.

So

any event you can think of will eventually happen, like existence of human life and all that sort of stuff.

But where that's not true that if it violates the laws of physics that yeah

these laws of physics theories like they really are

why why if if time is infinite if theoretically it's not but so let's say if it was infinite so you have a law such as the the conservation of electric charge which is based on some I don't have that law.

So you can't make a negative charge without a positive charge which is the way we think the universe works at the moment.

So that's why you can only create matter and antimatter in equal amounts, because you need to, if you're going to make some matter with a positive charge, you need to make an equal amount with a negative charge.

So that would be an absolute law.

Then no matter how long you wait for it.

Yeah, sorry.

No, you're absolutely right.

A physically impossible thing won't happen if it's physically impossible.

If there's a possibility that it's not impossible, then that will happen.

But I guess what I'm saying is all possible events will happen over enough time.

Yeah.

Yeah.

That's yeah.

Precision.

Bloody businesses, So can we hang on?

Can we get back to the man?

Sister's radio for it's about precision.

The listeners won't know this, but when Brian was explaining antimatter and matter, he was using it both with his fists

as if it were if lock stock and two smoking barrels had been made by the Open University.

You would have been a character in it.

We've got matter over here,

antimatter over here, and someone, I think, is about to go from matter to antimatter.

Thank you, Tim.

I am a physicist, you are a minstrel.

We can move on.

I'm going butch for the third series.

Right now, before we have many more questions, and of course, we have to deal with the.

What was it we were going to deal with after the next bit?

I can't remember.

Birthday paradigm.

Thanks, Tim.

That was a really handy reminder.

Now, I didn't think we would need our regular stand-up mathematician, Matt Parker, on the show, as we already have enough maths, as you can see.

But then Matt told me that that would lead to an increase of 37% in my likelihood of being attacked by a rook due to the decreasing number of people on stage.

I don't know that much about rooks, and I'm only just beginning to understand probability.

So, for that reason, here's Matt Parker.

I was on a bus the other day, and someone got on, and they were wearing exactly the same t-shirt as me.

It was awkward.

I thought one of us is going to have to say something.

He turned to me and just went, Oh, what are the chances?

Well,

because we can, We could work this out, right?

We need to know the density distribution of t-search in the population.

We could estimate the average frequency wear rate.

You look at the number of people you're close enough to each day to score a match.

And if you put all these together, you can work out if our matching Ts are significant.

It's the so-called statistical T's test.

When I did that in the maths department, it went down a treat.

We all agreed it was over 95% hilarious so who's the outlier now

but we can you can work this out and if you actually go through the numbers given the sheer number of people you come across each day I think it'd be more amazing if you never bumped into someone wearing the same thing as you it's like the media coverage last year of Wang Yang's marriage Wang Yang, who lives in China, married his fiancé whose name is also Wang Yang.

They've got exactly the same name and they were both born on the 29th of April 1982.

Identical birthday, identical names.

It seems amazing.

But you can actually look up the statistics on the number of names used in China, which I did.

And Yang is actually the sixth most common name in China.

There are literally millions of yangs in China.

Which is a sentence that gets far more racist the less context it gets.

To quote a taxi driver, it's not racist, it's a fact.

And Wang is actually the second most common name in China.

There are actually, at last count, count, 93 million Wangs in China.

And that's not racist.

That's an innuendo.

It actually turns out that half the Chinese population draw from just nine different names.

And if you want to factor in the same birthday,

the odds of a couple having the same birthday is one in 365.24.

You're right, sir, the precise number of days in the year.

And you can allow for years and you can actually work it out.

And it turns out, most coincidences, if you actually crunch the numbers, become a lot less amazing.

It was at this point that he got off the bus.

It turns out, no matter how many people say, what are the chances, a statistically insignificant number of them actually want to know?

Thank you very much.

Lovely moment there, as you nearly morphed into Bernard Manning.

There.

I'm not saying that Wang Yang isn't a culturally frequent name in China.

But it is.

So that's Bernard Manning, isn't it?

Yeah, I said his name again.

Who's this coming?

Why?

It's my old friend Tycho Bry.

I've got a cold.

I don't know where it comes out from now.

But we have got Alex, it's your birthday today, isn't it?

Yes, it is.

Yeah, so I think you've got a song for Alex, haven't you?

Happy birthday,

Paradox.

That's it.

So we're actually going to do a birthday paradox, as you said.

Matt is going to go along.

Which row, have you chosen a row, Matt?

I'm going to start systematically in the corner here and zigzag my way backwards.

Brilliant.

Before Matt starts, Alex, can you give us a little bit of the background then?

What is the birthday paradox?

The birthday paradox says that in any group of 23 people, it's more likely than not that two people will share the same birthday.

And the reason why we call it a paradox, when it's not as mathematically watertight, is that that seems a ridiculously small amount of number and it's incredibly counterintuitive to think that you only need 23 people for two people to share the same birthday.

And you actually only need seven people to be more sure than not that two will have been born within a week of each other.

And I think that's what we're going to try and find out.

I mean, it does sound that that is counterintuitive.

Well, there's a pretty damn good chance it'll go wrong after we

statistically.

Well, let's try this experiment then.

So, what we're going to do is we're going to pick a random row of the audience, which happens to be the front row over there because it's easier to get to.

And if you just say, you don't need to say the year of your birth, but if you just say the month and day of your birth, and let's see if we can get to two having a birthday within a week of each other.

Okay.

22nd of October.

19th of September.

September the 8th.

Oh, we're just about there.

That's right, yes.

No, we're not there.

And it doesn't work if you say we're just about there.

It wasn't a week, was it?

There was a

11 days.

Or was it 11 days?

Yeah.

I'm not.

Do you know what?

I'm beginning to think this Higgs boson thing's a little bit further off than we imagined.

He's terrible when numbers get under a trillion.

Carry on.

Within an order of magnitude is not enough for now.

26th of April.

8th of February.

2nd of June.

16th of October.

Let's double-check our prairie.

October date and 22nd of October.

Yeah.

And how many of that?

Let Brian work it up.

5, 6, 7.

Exactly.

Oh.

What a pity so many people hoping for a rollover with the birthday paradox this week on Radio 4.

So it was exactly right.

It's perfect.

Who are you clapping, by the way?

But if we did another group, surely it shouldn't work because there's only a 50% chance.

Oh, come on, we might as well.

No, but you're right.

It was very interesting, actually, because it does seem too good to be true, doesn't it?

And that's the thing about statistics, but I suppose it clearly works.

But go ahead, let's try it again.

Take two.

23rd of January.

29th of July.

18th of October.

5th of February.

2nd of February.

There we go.

Can I just say, Alex, this is the best birthday you've ever had, isn't it?

But likewise, if we were to say a date, I think if there are only about 200 people here, it's probably most likely that someone else doesn't doesn't share that birthday.

Oh,

so if the birthday becomes an observed birthday, then we're the act of observing it the odds on any two birthdays, not the odd that the odds aren't great, aren't absolutely brilliant that someone will have your birthday.

Yeah, you need, I think, just over 200 people to be pretty sure that someone will have your birthday.

Anyone to celebrate their birthday?

That's not

no one else's birthday is today.

Isn't it sad?

The only person who's actually turned up on their birthday for the show,

anyone else found something better to do.

So there's bias.

There's definitely bias there.

See,

again, that is a fascinating thing.

And people are, I hope, were reasonably interested in the fact that in such a small sample, you get that thing within a week.

Now, by understanding probability, if people truly understood probability, would the whole world of gambling collapse?

Or would it merely mean that everyone would then go, ah.

I have a system.

Because you hear with gamblers, they have a system.

The system is normally going home to their wife and saying, I don't know what happened.

I think I was mugged.

Yeah, definitely.

People would stop gambling.

I've interviewed lots of mathematicians, and none of them have said, Yeah, I love gambling.

They just don't do it.

I mean, what's the point?

Well, the only one who likes gambling, but this isn't really gambling, is he gambles on the lottery, but only when he can buy every single ticket.

I think, wasn't there some research done into this by people who make slot machines about what the payout rates had to be?

Because you know, I think everyone who plays a slot machine knows that if they carry on playing it, they'll lose.

But

there has to be an incentive to play, doesn't there?

The incentive is this sort of delusion called the gambler's fallacy, which is just say we know we're going to get a jackpot, say, one in ten, and we don't get the jackpot after nine, then we think, well, it's not happened nine, we're going to get it now.

But that doesn't work because every gamble is totally random.

But as human beings, we have this memory of what came before.

So we think, and that's the fallacy, oh, it's ready to give now, it's ready to give, but it's never ready to give, or it's always ready to give, it's always exactly the same probability.

But when we're playing, we tend to believe that there's a pattern behind it, and that forces us to carry on gambling.

We've evolved to see patterns where there aren't.

True.

Especially in vision and stuff.

If you're a Neolithic man in a jungle or woman in Savannah,

I don't want to be sexist or geographist.

You know, you're sitting there and you see a little bit of orange and a little bit of black and some grass and something.

It's worthwhile having the sort of brain that goes, tiger, out of not much data.

We see patterns as much as we possibly can.

It's interesting, isn't it?

I think what we're saying is that trying to behave rationally is almost irrational.

Well, yeah, so counter-instinctual, I suppose.

Well, totally, there's the interesting anecdote about the shuffle feature on the iPod.

People were getting the shuffle feature and they're saying, hey, but it just plays songs from the same album.

I have like a thousand songs.

How come it's always playing from the same album?

Well, you would expect clusters to happen, you know birthdays to happen together a shuffle to choose things together so people complain so Steve Jobs said I'm gonna make and he did he made the iPod shuffle less random so people thought it was more random

the interesting thing about the gamblers fallacy and all that to me is that um we're sort of talking as if it's a lack of understanding or knowledge of probability that makes people behave like this and that that's a factor but the reason people don't know about it is they don't seek that knowledge because they don't, in the first place, believe that they are the victims of chance in the world.

They actually believe that Caesar is choosing Caesar's head.

Most people in the world believe that their behaviours and the things they say influence the universe.

Well, it's built into our language, isn't it?

You hear it, it's meant to be.

I can't believe it, it's fate.

You know, I'm sorry to be a bit so optistic.

I am writing a musical of Matilda, and 10 years ago, I wrote to the Dalai State to ask for the rights of Matilda because I thought it was a great idea.

And then 10 years later, the RSC said, Do you want to write this song?

Totally coincidental.

And it's very hard to talk about that without engaging in the idea that it's the coming together of a fatalistic, you know, which is absolutely absurd.

And the self-importance you'd have to embrace to think that the universe is somehow over that 10 years, God, or someone's messing up there going, now, no, give it another couple of years.

The Dala state's not quite ripe.

I mean, how absurd.

And yet the instinct is is massive, isn't it?

To say, oh, we met.

What are the odds that we were going to meet that night?

If you hadn't have done that and I hadn't done that.

And that's true.

It's extremely low odds.

But our instinct is to explain it by saying, well, it's fate.

It's meant to be.

And I suppose this is also politically that there's a risk perception is very important, isn't it?

Let's say that you want to make the argument that nuclear power is safer than coal-fired power stations.

That understanding the risk

associated with potentially catastrophic events are flying, as you said, Tim, it's a common phobia when it's actually statistically

extremely safe.

So, is it worth trying to overcome that kind of irrational fear?

Because it is important if you're talking about whether to build nuclear power stations or whether to fly on planes or cars or trains.

Yeah, I think it's really important.

I mean, it's the most crucial thing.

I mean, it's so, so important.

And if you're wondering, nuclear power is much, much, much safer than coal power.

Just to do the most.

Often people say, What is it about mass that people should learn

that will really help them in your lives?

And I I don't think it is the ability to count or to calculate because we can use a calculator for that.

I think it is the ability to understand randomness.

And that brings us to the English on this.

I'll give you some of these as well, Brian.

These are the favourite numbers of the audience.

We ask the audience, what's your favourite number and why?

We've got a very broad range of, I would say, some very fine nerdery, some definitely interested mathematics, and some just absurdist, I think, would be the polite way of

saying that.

Just seven, because it's my favourite film, and I like to think that's how I'd live my life.

What?

Minus one, because I couldn't decide between E, I, and Pi, so I thought I'd go for E to the I, pie, which is superb.

That's quality, audience quality then.

664, The Neighbour of the Beast, now.

What?

What I particularly like about that is freak, I have heard that joke before, but it's been 665 or 667.

But Jason Seymour very cleverly thought, hang on a minute, that's not how it works with neighbours.

You have odds on one side and evens on the other.

So well done for finally, let's have it having a numerically accurate version of that joke.

In hell, all straights are triangular, so it actually skips three.

Eight.

Because it is well balanced and symmetrical, and when laid on its side, has infinite possibilities.

Alex, I know when we said we'd uh collected these earlier, you said you wanted to take them away afterwards to to to observe them.

I'm interested in numbers and also cultural differences and approaches to numbers.

And there's no reason why we should have a favourite number, but we all do, and so I'm kind of fascinated as to why.

Tim, do you have a favourite number?

I mean, if I had to choose a number, it'd be seven, because my wife and I are both born on the seventh, and it's a lucky number, and it's

not say

So, the chances of that are what?

About one in 250.

Oh, yeah, because it's very special.

So, it's not that special.

I've never claimed it was special.

So, that is the end of the show.

Hopefully, at the end of the show, you know more than you did at the start, but of course, there is the possibility that, in fact, you've learnt how much you didn't know that you didn't know, and therefore, the pie chart of what you know is smaller than it was 27 minutes ago.

Sorry about that, but we're now going to give you 167 and a half hours to do your revision.

Thank you to this week's guest, Tim Minchin, Alex Bellis, and Matt Parker.

Next week, we'll be joined by Alexis Sale, and we'll be discussing just how ridiculous philosophy is.

That's not obviously what we'll be talking about.

We will be discussing philosophy and its importance in culture, I think.

Look at you living in Plato's cave with all your shadows of whatever they are, things that shoot out magic.

And I recommend Plato's Republic, a quick flick through Tractatus Logico-Philosophicus and a little bit of the cartoon guide to Liebenit, which I'll be lending to you, Brian Cox.

And if you find those too difficult, may I recommend Peanuts with Charlie Brown's Snoopy.

Which is actually genuinely very, very existential, but you think Calvin Hobbes is more existential.

Yeah, but I haven't read any Charlie Brown.

It is a more difficult text.

I would agree.

Definitely.

Goodbye.