Does the average teacher really earn £54,000?

Summary

More or Less (BBC Radio 4)

Episode: Does the average teacher really earn £54,000?

Date: July 8, 2026
Host: Tim Harford

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Episode Overview

This episode of More or Less unpacks recent claims about teacher salaries in England, particularly the assertion that the “average teacher now earns £54,000.” The team digs into the realities behind the salary statistics, explores why such numbers can seem surprising or misleading, and then shifts to examine questionable claims about Russian soldier life expectancy at the Ukrainian frontline. The episode also tackles why plane travel can be cheaper than train travel in the UK and concludes with a deep dive into the math of card shuffling — and why every shuffle is likely unique.

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Key Discussion Points

1. Teacher Pay: Is the Average Really £54,000?

[00:48 - 05:59]

Context & Listener Reactions

• Baroness Jackie Smith claims the average teacher pay is “round about the £54,000 mark,” prompting disbelief from listeners.
• Multiple teachers and families report not recognizing this figure.
  • Quote, Luke Sibieta:

    "I screamed at the radio, I’m a top of pay scale teacher. The average is nowhere near £54,000." [00:57]

Breaking Down the Numbers with Luke Sibieta (Institute for Fiscal Studies)

• Current Median Pay:

  • Median (not mean) for classroom teachers in England is ~£51,000.
  • Upcoming pay rises could bring the figure closer to £54,000 in future years.
    • “Teacher pay is going to be around £54,000. So it's going in that direction.” [01:24]

• Why Is the Median So High?

  • About a third of classroom teachers are at the top of the upper pay scale (~£51,000).
  • Additional factors such as department responsibilities and London “weighting” (higher pay for London teachers) modestly increase the median but do not solely account for it.

• Regional Variations:

  • Top pay in London: ~£63,000 vs. ~£51,000 elsewhere.
  • However, median remains fairly representative nationwide.
    • “If you look at the mode teacher across England, they’re probably outside London, they’re probably at the top of the upper pay scale.” [03:16]

• Why Listeners Are Surprised

  • Teacher pay fell by 12% in real terms between 2010–2022; only recently have above-inflation pay increases begun to recover pay levels.
  • Despite gains, teacher pay is still ~7% lower in real terms than 2010.
    • “So they've seen about a 5% rise over the last few years and in the coming years.” [03:57]

• Relative to Other Professions

  • Teacher pay has slipped from the 80th-90th percentile of national incomes decades ago to “somewhere around the 60th or 70th percentile” now — above UK median pay, but not high compared to other professionals.

• Pensions

  • Teachers' pensions are more generous than most:
    • “If you look at the average employer contribution to pensions in the public sector, it’s currently around 29% for teachers. That compares with maybe around 6% in the private sector.” [05:19]

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2. Life Expectancy of Russian Recruits in Ukraine

[06:57 - 14:25]

Questionable Statistics Circulating

• Claims surfaced in The Independent and Foreign Policy:
  • “Russian soldiers surviving average of 20 minutes when they reach frontline…” [06:57]
  • Life expectancy from training to death: “10 days to three weeks.”

Source Investigation with Josh McMinn & Andrei Vladov

• Tracing the Stat:

  • Originates from small Russian Telegram channels, notably “House among the Laurels.”
  • The cited study is untraceable and possibly fabricated; referenced by Telegram bloggers, republished by Astra, then Foreign Policy, and The Independent.
    • “Wait, wait, wait. So you’re telling me that the source of this statistic is a study that an unknown Telegram user says they remembered seeing once, and that this was then reported by a Russian social media channel, which was then reported in Foreign Policy? Which was then reported in the Independent?” [09:15]

• Reliability:

  • The source lacks credibility; not widely regarded among Russian pro-war bloggers.

Is It Plausible? With Mark Galeotti (Author, Russia Military Historian)

• Statistical Impossibility:

  • Actual numbers of Russian dead/wounded are far below what the 20-minute or three-week statistic would require.
    • “Even if we take the most generous assessment... the figures just don't add up.” [10:05]
  • The structure of modern drone warfare involves infiltration teams and protracted operations, not mass charges with instant huge loss rates.

• Nuance:

  • While many Russian recruits die quickly, broad claims of 20-minute average survival cannot be substantiated.
  • “If we regarded that as a figure across the board, then the Russians would be having massively greater casualties than everyone, even the most optimistic Ukrainian accounts are claiming.” [14:03]

• What Does This Say?

  • Even unverifiable rumors being discussed among pro-war Russian bloggers signals acute awareness of high casualties and drones' lethality on both sides.

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3. Is It Cheaper to Fly or Take the Train?

[16:15 - 21:12]

Price Comparison with Simon Calder (Travel Journalist)

• Testing a Classic Route: London ↔ Edinburgh

  • Direct airline competitors: BA, EasyJet, Ryanair
  • Direct train operators: LNER, Lumo, Avanti West Coast

• Findings:

  • Next-day travel: Cheapest train is ~£80 (evening), cheapest plane at similar time is £138, but drops to £79 later.
  • Four weeks ahead: Trains £50+, planes £30+; sometimes Ryanair flights as low as £15, while the cheapest train is £42.

• Factors Influencing Price:

  • Occupancy:
    • “In general, one in three seats filled on the average train… airlines fill up all their seats.” [18:52]
  • Rail Subsidy & Costs:
    • Trains receive significant taxpayer support; maintenance of old railways (“£400 per second”) weighs heavily.
  • Competition:
    • Where train routes have no rail competitors (Eurostar to Paris/Brussels), “they exploit the market,” charging much higher fares than airlines.
      • “I honestly can't remember the last time… I took the train from London to either Paris or Brussels because… I’m always looking at £200 one way. And the airlines are always saying 60, 70, £80, those sorts of prices.” [20:33]

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4. Card-Shuffling Math: Is Every Shuffle Unique?

[21:43 - 26:50]

Expert: Matt Parker (Mathematician/YouTuber)

• The Claim: Every shuffle of a 52-card deck yields a unique, never-before-seen order.
• Mathematical Reasoning:
  • The number of possible arrangements is 52 factorial (52!), or 8 followed by 67 digits.
  • For 13 cards, possible arrangements already exceed 6 billion.
• Probability:
  • Even if “10 billion humans shuffled once a second for longer than cards have existed,” the odds of any two shuffles matching are effectively zero.
    • “You still end up with an outrageously big number… a chance of 1 in 4 x 10^26.” [26:09]
  • “Not strictly zero… I would say it’s indistinguishable from zero. Or it's effectively zero.” [26:26]

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Notable Quotes & Memorable Moments

• On teacher pay perceptions:

  "I screamed at the radio, I’m a top of pay scale teacher. The average is nowhere near £54,000."
  — Luke Sibieta [00:57]

• On the source of questionable war stats:

  "Wait, wait, wait. So you’re telling me that the source of this statistic is a study that an unknown Telegram user says they remembered seeing once…?"
  — Tim Harford [09:15]

• On rail subsidies:

  “Last time I calculated it was £400 per second. You are talking billions upon billions of subsidies.”
  — Simon Calder [19:44]

• On card shuffles and mathematical scale:

  “You still end up with an outrageously big number. However, not strictly zero. And this is why people get upset when I say it definitely hasn't happened… I would say it's indistinguishable from zero.”
  — Matt Parker [26:09, 26:26]

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Timestamps for Key Segments

• Teacher salary claim unpacked: 00:48 – 05:59
• Russian frontline survival stat debunked: 06:57 – 14:25
• Plane vs. train travel cost explained: 16:15 – 21:12
• Card shuffling & combinatorics math: 21:43 – 26:50

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Tone and Style

The episode is characterized by More or Less’s signature tone: curious, analytic, gently humorous, and scrupulously fact-checked. The host and guests break down complex statistics in plain English, often using relatable analogies, and are quick to challenge dubious claims—while always maintaining skepticism and a dash of fun.

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Summary Takeaway

More or Less delivers an accessible breakdown of how headline stats often mislead or oversimplify — whether about teacher pay, war casualties, travel costs, or shuffling cards. Behind every remarkable-sounding claim, there’s nuance: real teacher salary medians, social context for war stats, market economics for trains and planes, and the mind-boggling scale of combinatorial math.

Transcript

[Tim Harford]

Hello and welcome to More or Less. We're your weekly guide to the numbers all around us, from Manchester Piccadilly to the Finland station. And I'm Tim Harford. This week, what is the life expectancy of a Russian recruit on the Ukrainian front line? And how on earth would we know? Is it cheaper to fly or take the train? And why? Ant Maths YouTuber Matt Parker claims that every shuffle of a deck of cards is an unprecedented event. Every ace or joker. But first, last week, Baroness Jackie Smith, Minister for Skills in the Department for Education, was on Radio 4's Today programme. She was asked about a recent row over the government offer on teachers pay in England. During the discussion, she told listeners, that's

[Matt Parker]

the average pay of a teacher now,

[Mark Galeotti]

round about the 54,000 pound mark.

[Tim Harford]

Several listeners heard this and had their doubts.

[Luke Sibieta]

I screamed at the radio, I'm a

[Tim Harford]

top of pay scale teacher. The average is nowhere near £54,000.

[Luke Sibieta]

Can you please investigate this for us

[Mark Galeotti]

as we have three qualified teachers in

[Luke Sibieta]

our family who do not recognise this figure?

[Tim Harford]

Is the average pay of a teacher around £54,000? We asked Luke Sibieta, an expert on schools and education funding at the Institute for Fiscal Studies.

[Luke Sibieta]

The average teacher pay is going to be around £54,000. The median pay for teachers in England is currently around £51,000. But if we add in the pay rises that are going to be happening this September and next September, then teacher pay is going to be around £54,000. So it's going in that direction.

[Tim Harford]

Okay, so I think we can forgive a slight misspeak. We've all done it. Our listeners are clearly surprised that the number is that high and would like it to be explained to them. How is that number calculated? We should start. Is it median or mean? And what does the distribution of teacher salaries look like?

[Luke Sibieta]

So this is the median pay amongst all classroom teachers. So teacher pay is structured according to pay scales going from the bottom of the main pay scale up until the top of the upper pay scale. But teachers move through these pay scales relatively quickly. And about a third of all classroom teachers are at the top of the upper pay scale, which is around £51,000, which is the median across all teachers.

[Tim Harford]

So how is it possible that the median classroom salary is the same as the salary that you would earn if you're in the very top bracket as a teacher?

[Luke Sibieta]

So this probably all comes from the fact that there's a very, very large number of teachers on the upper pay scale. So there's around a third of teachers across the country who are on the top of the pay scale. That's not enough, obviously, to take that to the median point. But there's Also around over 15, 20% of teachers are on just below the upper pay scale. And there will be some teachers who are receiving additional payments for being head of year group or head of department. There'll be some teachers receiving London allowances as well, which is why the median is around £51,000.

[Tim Harford]

And is this being driven by the London weighting that teachers in the London area get?

[Luke Sibieta]

So teachers in London do receive high levels of salary. So for example, in London, the top of the pay scale is around 63,000 pounds as compared with around around 51,000 pounds outside London. So that will be pushing up the median a little bit, but the differences aren't big enough to be making a huge difference. An average salary level at £51,000 is about the norm across the country. If you look at the mode teacher across England, they're probably outside London, they're probably at the top of the upper pay scale. 51,000 is a fair reflection of the average teacher pay level across England.

[Tim Harford]

Why do you think it is that so many of our listeners find these numbers surprising?

[Luke Sibieta]

It might come from the fact that teachers have seen real terms pay cuts over the last 15 years. So between 2010 and 2022, teacher salary levels fell by around 12% in real terms. They've now been creeping up over the last few years with Pay rises around 5.5%, 4% and above inflation rises planned for the next few years. And as a result of those changes, teacher Pay is around 7% lower than it was in 2010. So they've seen about a 5% rise over the last few years and in the coming years.

[Tim Harford]

I think it's so interesting that over the last 15 years or so, teacher salaries have dropped in real terms, and yet many of our listeners are so surprised about how high they are.

[Luke Sibieta]

Historically, teacher pay levels have been higher and further up towards the top of the income distribution. So if you were to look back into say the 80s or the 90s, you'd see teacher pay teacher salary levels further towards around the 80th or the 90th percentile of the income distribution. So in relative terms, teacher pay has fallen as compared with other professional occupations over the last 20, 30 years. If you look at all salaries across the economy, teachers are somewhere around the 60th or 70th percentile. So they're well above the median, but not towards the upper echelons in the 80th 90th percentiles.

[Tim Harford]

What about teacher pensions? Are they regarded as being quite enviable or not particularly so.

[Luke Sibieta]

If you look at teachers pensions, they're very generous and they're an example of a public sector pension. And public sector workers tend to receive relatively high levels of pension payments. They're in the form of defined benefit schemes, which is very unusual in the private sector, but they're also high in terms of their value and contribution rate. So if you look at the average employer contribution to pensions in the public sector, it's currently around 29% for teachers. So they get the equivalent of around 29% employer pension contributions. That compares with maybe around 6% in the private sector. So teachers pensions are very generous indeed as compared with other occupations.

[Tim Harford]

Our thanks to Luke Sibieta from the Institute for Fiscal Studies. You're listening to More or Less. Great decision. And why not repeat that behaviour by binging our snack sized Saturday podcasts. Last week we asked whether full moons make CR crime rates go up this Saturday. Does playing tennis extend your Life by nearly 10 years? Go on, feed that impulse like a rat in a cage. Your next dopamine soaked dose of statistical scrutiny is just a few clicks away. Last week the Independent published the headline russian soldiers surviving average of 20 minutes when they reach frontline due to drone attacks. 20 minutes is not a very long time. Can that possibly be true? Our resident data mole, Josh McMinn, has been digging down to find the source of this stat. Hello, Josh.

[Josh McMinn]

Hi Tim.

[Tim Harford]

So where did the Independent get this stat from?

[Josh McMinn]

Well, the Independent cites an article by Oxford historian and author Professor Peter Frankpan in the magazine Foreign Policy. That article has yet another surprising life expectancy station.

[Mark Galeotti]

Here's what it According to Russian military bloggers, the average life expectancy of a new recruit, from arrival at a training ground to death in a combat zone, lies somewhere between 10 days and three weeks. Once they're sent onto the battlefield, Russian fighters survive an average of 20 to 35 minutes.

[Tim Harford]

Hmm, 10 days to three weeks. 20 to 35 minutes. The claim's already gotten a little mushy. So who are these Russian military bloggers that Peter Frankman talks about?

[Josh McMinn]

So these are accounts on an app called Telegram, which is a bit like WhatsApp, but where the chats can have over a million members. People follow Telegram channels to get fast information on what's happening in the war. The channel cited in this article is a Russian investigative news source called Astra.

[Tim Harford]

And what did Astra find?

[Josh McMinn]

I thought Astra might be the source of this statistic, but they're not. Astra referenced an even smaller Telegram channel called House among the Laurels.

[Tim Harford]

Oh, so we're quite deep down the rabbit hole now.

[Josh McMinn]

And it only goes deeper. Even House among the Laurels aren't the source of this statistic. In their post, they say they remember seeing the numbers in a study by someone simply known as N. Sounds rock solid.

[Tim Harford]

So do we know who this N character is?

[Josh McMinn]

Well, no. This is where I recruited the help of some real Telegram experts to help me figure this all out. Here's Andrei Vladov from BBC Monitoring.

[Andrei Vladov]

Actually, the House among the Laurels is a very small channel. It has only got about 5,000 subscribers, so I wouldn't say it's an influential one. We did quite an extensive search using a Telegram monitoring tool that we have, and it shows that the post was shared only by a few channels in May, and the largest of those was an anti Kremlin channel that watches the pro war blogosphere, which is to say that it did not create much of an impression among pro war bloggers. We were also not able to find posts that refer to the same study, which, for all we know, may have been just made up by this anonymous author.

[Tim Harford]

Wait, wait, wait. So you're telling me that the source of this statistic is a study that an unknown Telegram user says they remembered seeing once, and that this was then reported by a Russian social media channel, which was then reported in Foreign Policy? Which was then reported in the Independent?

[Josh McMinn]

Yep.

[Tim Harford]

Oh, dear. So it's clear that this surprising claim originates from a less than reputable source. But could it still be true? To find out more, I spoke to Mark Galeotti, author of Forged in A Military History of Russia From Its Beginnings to Today. So let us deal with the life expectancy of a new recruitment. Is it plausible that new recruits only last from 10 days to three weeks from their training?

[Mark Galeotti]

I mean, it really isn't. First of all, I mean, even though the training is shockingly little, they generally will get at least a week's training. Then they have to move to their unit before they can be thrown into any particular meat grinders. More to the point, look, there's about 700,000. A little bit more than that. Russian troops currently engaged in the war as a whole. Now, of course, the overwhelming majority of those are not involved in assaults at any one time, but there's probably about 50,000 who are. Now, even if we take the high end of that three weeks, that would mean that their casualties, their dead, would be substantially more than 50,000 every month. And in fact the figures we have are, roughly speaking, 35,000 dead and wounded, with the very highest estimates of actual dead being 20,000. Now, as I said, so even if we take the most generous assessment of these claims from a Z blogger and also the highest end assessments of Russian casualties, the figures just don't add up.

[Tim Harford]

What about this headline in the Independent that Russian soldiers are surviving an average of 20 minutes when they reach the front line and that's because of drone attacks? What do we make of that?

[Mark Galeotti]

I've been trying to twist myself into various knots, trying to work out a way of explaining how that could be the case.

[Tim Harford]

That's very generous of you because when I heard it I just thought, well, that cannot possibly be true.

[Mark Galeotti]

Well, it cannot possibly be true. But in a way there's so much we don't know about this war that one should be very careful about discarding any data points, even when it comes from a very, very questionable source. But the point is, it can't possibly be true as you assumed. I mean, we have to remember after all that the Russians, however painfully, are still making advances. According to the most recent estimates in the last four weeks, they took, in aggregate terms, territory about the size of the Isle of Manhattan. There's no way they could be doing that, frankly, and sustaining this level of losses.

[Tim Harford]

Yes, I mean, I suppose one way of looking at it, and it's not really my job or your job to try to twist ourselves into a position where the number could make sense, but maybe if we use only the men who have died and not the ones who survive, maybe it's true that of the soldiers who have died, they have typically died within half an hour. Could that be true?

[Mark Galeotti]

The truth of the matter is that actually we now have a rather bizarre kind of war being fought. It's not like the Second World War. It's not like the First World War. The drone war is one that is being fought by the Russians sending very small groups of infiltrators, two, three man teams that try and sift their way through the drone scoured front lines or area, hoping not to be spotted, because to be spotted is to be killed and then go to ground somewhere behind the front line in due course to meet up with other little groups of infiltrators until you've got a force that is just about large enough to maybe assault a Ukrainian position. So in other words, this is something that doesn't take place very quickly. It's not like Again, World War I, an officer blows a whistle and all the poor riflemen boil out of the trenches to be gunned down by machine guns. This is one that can often take. Actually, the conflict takes place on a timeframe, often of days rather than minutes. Now, maybe what that means is that perhaps the green recruits who are going to die maybe die in the very early minutes of the war, whereas the seasoned veterans who understand the rules of the game, last a lot longer.

[Tim Harford]

So, stepping back, we've got this claim that soldiers are dying within 10 days, three weeks of basic training, we've got this claim that soldiers are dying within 20 minutes or so of entering combat. Are either of these claims at all plausible?

[Mark Galeotti]

I mean, the top line answer is not really. It may well refer to certain specific classes of soldiers, particular times of engagement or whatever, but if we regarded that as a figure across the board, then the Russians would be having massively greater casualties than everyone, even the most optimistic Ukrainian accounts are claiming.

[Tim Harford]

While it's undoubtedly true that Russia is throwing thousands of barely trained soldiers into a deadly drone kill zone, many dying very quickly, you should take these specific claims of survival times with a pinch of salt. When we asked Peter Frankopan about this, he agreed that you can't verify the claims, which is why he sourced them to the Russian bloggers. But he told us that the fact that pro war bloggers are even having these conversations tells you something interesting. Our thanks to Mark Galeotti, author of Forged in War. We made a terrible mistake on a recent episode. We answered a question from the depths of our inbox, and this, although we did not realise it at the time, risked encouraging. Les autres. Loyal listener Julian got in touch. This has emboldened me to ask again a question that I asked you in September 2023, but I think has become no less relevant in the meantime. Why is air travel cheaper than rail travel? That's the question. Just the kind of enormous question with a potentially highly ambiguous answer that we do our best to avoid on More or Less. But there is one person we trust to have a reasonable go at it, as BBC producers have done through the ages. At times of travel uncertainty, it is time to summon Simon Calder, no longer the Independence travel correspondent, but now of the Telegraph, where he's the presenter of the new podcast, the Travel Expert. Okay, here goes. Simon Calder. Simon Calder. Simon Calder.

[Simon Calder]

Hello.

[Tim Harford]

So, Simon, welcome back to More or Less. I should first ask, are you speaking to us from Himalayan Plateau?

[Simon Calder]

I'm at Manchester Piccadilly Station, one of the very busiest stations outside London.

[Tim Harford]

Well, I feel that you've already picked a side by choosing to speak to us from Manchester Piccadilly rather than, for example, Manchester International Airport. But okay, fine. So look, our loyal listener is very keen to find out why rail travel is more expensive than air travel. And I suppose the first thing we should do is to ask, well, is it?

[Simon Calder]

That all depends. Let me take what I think is a fair comparison, because it is the busiest city pair for airlines in the uk and that is going from London to Edinburgh or indeed back again. And that is, first of all available through three different airlines, British Airways, EasyJet and Ryanair. And it's also accessible on LNER, which is state run, Lumo, which is a private operator, and Avanti West Coast.

[Tim Harford]

We're going to avoid the question of luggage here, mainly because I find it intensely triggering. So, with the playing field set, who wins a price comparison?

[Simon Calder]

If I'm looking a day ahead for the cheapest train fare, I'm looking at about £80, and this is for the evening rush hour. And if you go to the planes, then the cheapest fare at actual tee time is 138 pounds. But if you fly at 9:20 in the evening, and frankly, because of the speed of planes, you catch up with the 6:30 train out of King's Cross, that falls to £79. So that's fairly equivalent. And looking four weeks ahead, it's a different picture. Cheapest train about £50. The cheapest plane about £30. If you're super affordable and looking for the very lowest price, there's plenty of £15 flights on Ryanair. Looking everywhere for the cheapest train I could see. £42 was the answer.

[Tim Harford]

Okay, so if you have Simon Calder level travel planning skills, then the difference in cost between rail and plane on a journey like this isn't particularly big. Still, though, intuitively, it feels like a plane that you have to launch through the sky shouldn't be competitive with a train that you roll along tracks with a lot more seats on it. So what's going on? The first thing to recognize is how many of those seats actually have people sitting on them.

[Simon Calder]

In general, one in three seats filled on the average train. And that means, unfortunately, you are moving quite a lot of fresh air around and those empty seats aren't paying you any money.

[Tim Harford]

A fully private train operator such as Lumo just pays to use the tracks and can run their service for maximum profit, which means running trains when they're full. But most operators have fixed obligations to cover certain routes, even if the trains are mostly empty.

[Simon Calder]

Unlike the airlines, who average roughly 85% Ryanair is pretty close to 95% of filling up all their seats.

[Tim Harford]

Then there's the question of tax. Aviation fuel is exempt from fuel duty, but airlines do pay air passenger duty. However, if you're considering the overall fiscal cost of the Exchequer, rail gets a lot of public support.

[Simon Calder]

No, rail passengers in the UK pay VAT on their travel. In fact, quite the opposite. Many of them get a massive subsidy from long supply taxpayer. It all depends how you look at the costs of travel, whether you're including the very, very high costs of infrastructure. Not just building new railways, but maintaining the Victorian infrastructure. Last time I calculated it was £400 per second. You are talking billions upon billions of subsidies.

[Tim Harford]

So where does that leave us in answering Julian's question? Well, when there's competition on busy lines, rail doesn't do too badly compared to planes. Where there's no competition, however, economics takes its course.

[Simon Calder]

The only operator between London and Paris, Brussels, Amsterdam by train is Eurostar. Now, they exploit, as you would imagine as an economist, they would exploit the market. They limit the capacity. They have a lot of very expensive tickets for peak times, particularly if you're booking in advance. I honestly can't remember the last time, I'm sorry to say, I took the train from London to either Paris or Brussels because I normally book quite late and I'm always looking at £200 one way. And the airlines are always saying 60, 70, £80, those sorts of prices.

[Tim Harford]

Our thanks to the travel expert Simon Calder. Now, can I interest anyone in a game of cards? Contract Bridge, Texas hold', Em, Black Maria, PK German Whist, Snap here on More or less. We love a good card game, but it's not the gambler's thrill that keeps us coming back, or even the bragging rights. It's the fascinating world of probability that a single deck can conjure.

[Matt Parker]

Famously, if you shuffle a deck of cards, you will end up with an arrangement that's never been shuffled before. Because there are just so many ways you can arrange 52 cards, the probability of someone else having done it exactly the Same is basically zero.

[Tim Harford]

That is Matt Parker, a maths YouTuber and stand up comedian, talking on his podcast A Problem Squared. But can that possibly be right? Can just 52 cards be arranged in so many ways that it means every shuffle is unique? We invited Matt into the studio to show us his workings.

[Matt Parker]

Now, I didn't realise how bold a statement I was making at the time, but I'm here to defend it. In fact, I've brought in a deck of cards.

[Tim Harford]

Okay, Good.

[Matt Parker]

So I can shuffle them for you. And live on radio. This is just like a nice, lazy, nice lazy overhand shuffle.

[Tim Harford]

Yeah, I would expect better from you, Matt, to be honest.

[Matt Parker]

Now. Okay, we'll split the deck if, if I can get this to work. We'll do a. Oh, riffle shuffle.

[Tim Harford]

That's a mid air riffle shuffle, by the way, for those of you listening in audio only.

[Matt Parker]

So if people are happy, if I've done a couple of these, if we're happy to say that's a random arrangement.

[Tim Harford]

Yeah.

[Matt Parker]

No one has ever shuffled a deck of cards into this specific order before.

[Tim Harford]

World premiere, folks.

[Matt Parker]

Exactly. You heard it here first.

[Tim Harford]

Talk us through why you are so confident.

[Matt Parker]

If you want to work out the number of ways you can arrange a deck of cards, you're straying into the maths topic of combinatorics in terms of counting combinations. And specifically you're using what's called a factorial, which a lot of people remember as the little shouty exclamation mark on the calculator. Which is. Exactly. And it's such a perfect, like there's very little mass notation. Which is so perfect as using the exclamation mark for something as startling as factorials.

[Tim Harford]

Because factorials make exponentials look puny.

[Matt Parker]

Yes. They just get so ridiculously big so quickly. And I think what makes them extra startling is they kind of drop out of seemingly such boring, small contained situations.

[Tim Harford]

So let us return to the boring, small contained situation. How do factorials tell us something about a deck of cards?

[Matt Parker]

So if you just wanted to work out how many ways you could arrange one suit, let's say the 13 clubs.

[Josh McMinn]

Yeah.

[Matt Parker]

There are 13 cards you could put down first.

[Josh McMinn]

Yeah.

[Matt Parker]

And then there are 12 you could put down next.

[Tim Harford]

Yeah.

[Matt Parker]

And there are 11 you can put down after that. And each time you've got one fewer option because you've just put a card down. And so to work out, the total number of options you're presented with is 13 times 12, times 11 times 10, all the way down.

[Josh McMinn]

Yes.

[Matt Parker]

Which eventually multiplying by one.

[Tim Harford]

Do the full calculation and you find that a mere 13 cards can be shuffled into over 6 billion unique arrangements. And as factorials increase, they get bigger quickly. Alarmingly quickly. In fact, by the time you get to 52 factorial, which is the calculation you need for a full 52 card deck, the total number of possible arrangements

[Matt Parker]

is, well, it's eight and then another 67 digits after that.

[Josh McMinn]

Right.

[Matt Parker]

So we've kind of gone past giving numbers nice names.

[Tim Harford]

Yes.

[Matt Parker]

And we just say how many digits they've got.

[Tim Harford]

Billions, trillions, quadrillions, quintillions, septic. You're so far beyond that, forget it. So, Matt, you have convinced me that the number of different possible combinations of a deck of cards is very, very large.

[Matt Parker]

It's very big.

[Tim Harford]

Very, very, very, very large. Still, the claim that there is no chance. Yes.

[Matt Parker]

No, you're absolutely right. Now, I should say, because just having lots of combinations is one thing.

[Josh McMinn]

Yeah.

[Matt Parker]

But what we actually care about is have two people shuffled the same deck?

[Josh McMinn]

Yes.

[Matt Parker]

And now we're gonna look at every possible pairing of shuffles that may have happened.

[Tim Harford]

Okay. So every time someone has shuffled, they've created an arrangement, and then every subsequent shuffle might match with that or any other historical arrangement. And you've got a lot of chances to hit.

[Matt Parker]

A lot of chances to hit. You still end up with an outrageously big number. So even if you had 10 billion humans shuffling once a second for longer than cards have been invented, and then you run the combinations on, what's the odds any two of those matched, you're still looking at a chance of 1 in 4 times 10 to the 26.

[Josh McMinn]

Right.

[Matt Parker]

It's like outrageously big number. However, not strictly zero. And this is why people get upset when I say it definitely hasn't happened. Here's the thing. Because people get stuck on the fact it's not exactly zero.

[Josh McMinn]

Yes.

[Matt Parker]

I would say it's indistinguishable from zero. It's. Or it's effectively zero. That might be a safer way to put it. And there are shades of being basically zero. Because if you say something's one in a million. Yeah, that might still happen. We've got millions of people. If you say something's one in a billion or one in a trillion, at some point, we don't have enough opportunities in our universe for it to happen.

[Tim Harford]

Our thanks to friend of the programme, Matt Parker. You can find him on YouTube as standupmaths. And thanks also to friend of the programme, shuffler extraordinaire, Hugh Levinson. That's all we have time for this week, and indeed this series. Never fear, our shorter Saturday episodes are available all year round. And the podcast Cautionary Tales with Tim Harford is also available each week on BBC Sounds. Meanwhile, please keep your questions and comments coming in to more or lessbc.co.uk we'll be back in September, and until then, goodbye. More or Less was presented by me, Tim Harford. The producer was Tom Coles with Nathan Gower, Josh McMinn and Lizzie McNeil. The production coordinator was Siobhan Reed. The programme was recorded and mixed by James Beard. And our editor is Richard Varden.

[Luke Sibieta]

Hello.

[Greg Jenner]

Greg Jenner here. I'm the host of youf're Dead To Me, the BBC comedy show that takes history seriously. Every episode, I pair up a top historian with a fantastic comedian, and we have a lovely, funny, fascinating chat about a different subject from world history. And in this new series, we're beginning with an epic voyage through the story of Homer, the Iliad and the Odyssey. And that's with Kael Smith Bino joining us. And then we'll be learning about Francis Galton and the racist, discredited pseudoscience of eugenics. With Desiree Burch, we'll meet many medieval saints in their bone boxes. With Rachel Paris. That's yous're Dead To Me. Listen. First on BBC Sounds.