Using statistics in court

Summary

More or Less (BBC Radio 4)

Episode: Using statistics in court

Air date: January 6, 2012
Host: Tim Harford

---

Episode Overview

This episode of "More or Less" delves into the use—and misuse—of statistics in the courtroom, highlighting how mathematical reasoning can both illuminate and cloud legal outcomes. Tim Harford and his guests explore classic fallacies made with DNA and other forensic evidence, the courtroom debate around Bayes’ Theorem, and why mathematical logic still makes many legal professionals uneasy. The episode also addresses a popular listener marital dispute about lottery odds, the calculation of rail fare increases, and an unexpected foray into the infinities of numbers.

---

Key Discussion Points & Insights

---

1. Statistics and Probability in Courtrooms ([00:32]-[09:21])

Statistical Fallacies: The Prosecutor’s and Defendant’s Fallacy

• Angela Saini explains that both lawyers and forensic experts often get statistical reasoning wrong:
  • Defendant’s Fallacy: Example from the O.J. Simpson trial, where defense argued that few abused women are killed by their partners (<1 in 1,000), while statistics actually show that if a woman is abused and then murdered, there’s an 80% chance her partner did it ([01:34]).
  • Prosecutor’s Fallacy: When the rarity of a DNA profile match (e.g., 1 in 3 million) is conflated with the chance of a suspect’s innocence—ignoring the size of the suspect population, as in the Andrew Dean case ([02:37]):

    “The jury assumed that the chance of Andrew Dean being innocent must also be 1 in 3 million... In a population of 20 million adult men... you’d expect there to be around seven matches. And this is why, in the end, Andrew Dean’s conviction was quashed.” – Angela Saini ([02:37])

Bayes’ Theorem: The Logical—But Unloved—Solution

• Bayes’ Theorem is correct for updating how likely a suspect is guilty with each new piece of evidence ([03:43]).
• Problem: Courts are reluctant; some judges even ban its use, fearing it confuses jurors (e.g., 2011’s “R v. T” case banned Bayes for fibre evidence) ([04:03]).

  “Statisticians love numbers and they love Bayes, but judges don’t. Courts have even ruled in the past it formally confused jurors.” – Angela Saini ([04:03])

What Bayes’ Actually Means – Prof. David Spiegelhalter

• Bayes Theorem uses the likelihood ratio to weigh new evidence and update beliefs about guilt or innocence ([04:40]).
• Classic misuse: Sally Clark case (sudden infant deaths, wrongful conviction based on misunderstanding likelihood of rare events; failing to compare plausibility of guilt vs. innocence) ([05:44]):

  “It’s very easy to make that change in reasoning, which is completely wrong.” – Prof. David Spiegelhalter ([06:47])

• Fundamental misunderstanding: Probabilities are informed by experience and data, not hard facts. Banning probabilistic reasoning leaves juries with vague, misleading phrases ([08:40]):

  “It would be really a huge loss if we threw out the whole possibility of rational reasoning… just because we can’t agree on some numbers to three decimal places.” – Prof. David Spiegelhalter ([09:02])

---

2. Are UK Rail Fare Hikes as Bad as Reported? ([10:00]-[14:17])

Weighted Averages & Fare Comparisons

• The average rail fare increase (reported at 5.9%) may seem misleading; calculation depends on whether the average is weighted by route popularity or just ticket type ([10:00]).
  • Example: If busy route fares go up dramatically and quiet ones don’t, true revenue-weighted average is higher ([10:17]).
  • Charlotte MacDonald: Train companies and ATOC do, in fact, use a weighted average ([11:08]).

International Comparisons

• UK fares appear much higher than on the Continent—but key differences exist in frequency, speed, and the way public subsidies affect pricing ([11:44]).

  “In Britain, we have decided it is the rail user who should pay the higher portion of it.” – Prof. Roger Vickerman ([13:17])

• European fares are often lower, but systems differ in how much cost is borne by government and by travelers ([13:10]).

---

3. Numbers, Infinites, and Listener Corrections ([14:36]-[18:38])

Infinity of Primes

• Listener Paul Clark questions whether the sum of all primes is infinite; Matt Parker affirms it is:

  “Infinity is not an extreme. It’s something that just never ends… Infinity is not just a very, very big number.” – Matt Parker ([15:58])

• There are different sizes of infinity (countable and uncountable), with real numbers being a larger infinity than natural numbers or primes ([17:00]-[17:46]).

  “We know there are an infinite number of infinities.” – Matt Parker ([18:33])

The Size of Japanese Debt… in Units of Wales

• A listener calculates how much of Wales could be covered with £1 coins to represent Japan’s debt ([19:14]).
  • 1 quadrillion yen = 8.37 trillion pounds, enough to cover only 17.7% of Wales with £1 coins ([19:43]).

---

4. Lottery Odds, Patterns, and Marital Disputes ([20:56]-[27:16])

Are Consecutive Lottery Numbers More Likely than Random Ones?

• Listener Julie: Ongoing marital argument—are straight runs (1,2,3,4,5,6) more likely than random numbers? ([21:45])
• Prof. Kevin McConaughey: All six-number combinations are equally likely (about 1 in 13 million):

  “Any set of six numbers is just as likely to come out as any other set of six numbers.” ([22:14])

• Most people avoid straight sequences, but if you pick common patterns, you risk sharing your jackpot ([24:22]).

  “If you choose something that looks terribly random, or ideally perhaps use the system that allows the lottery to choose for you, that might be a better bet.” ([25:43])

• The only aspect that matters about which numbers you pick is whether other people are also picking them—i.e., it doesn’t change your odds, just your chance of having to share.

Statistically Significant Pattern in Listener Marital Disputes?

• Both listener disputes on probability have had the wife correct and the husband mistaken ([25:52]):

  “And a quarter is not terribly unlikely really. So it doesn’t really convince me of anything…” – Prof. Kevin McConaughey ([26:54])

• Ongoing invitation for listeners to submit their own disputes for adjudication.

---

Notable Quotes & Memorable Moments

• “[Bayes’ Theorem] is a formal way of using imperfect evidence to weigh up competing explanations… but you also have to take into account the plausibility of the explanations themselves based on other knowledge you've got.”
  — Prof. David Spiegelhalter ([04:40])

• “If you throw out the use of probabilistic reasoning, when there’s ever any question about what the probabilities are, you’re left with phrases such as, ‘Oh, this could have come from and is consistent with,’ which is… ripe to be misleading.”
  — Prof. David Spiegelhalter ([08:40])

• “Infinity is not just a very, very big number. Infinity is not an extreme. It’s something that just never ends.”
  — Matt Parker ([15:58])

• “Any set of six numbers is just as likely to come out as any other set of six numbers… The chance of getting 1,2,3,4,5,6 is minuscule… but the same goes for every other set of numbers as well.”
  — Prof. Kevin McConaughey ([22:14] and [23:14])

---

Important Segment Timestamps

• Statistical fallacies in court: [01:34]–[03:38]
• Bayes' Theorem in legal context: [03:43]–[09:21]
• Rail fare increase calculation: [10:00]–[14:17]
• Infinity of numbers and listener corrections: [14:36]–[19:14]
• Japanese debt in terms of Wales: [19:14]–[20:36]
• Lottery odds marital dispute and explanation: [20:56]–[26:16]

---

Conclusion

This episode masterfully demonstrates why statistical logic matters deeply in courts—but also why it can terrify lawyers and judges. Listeners also learn when average calculations mislead, that all six-number lottery tickets are equally (im)probable, and get a quirky lesson in comparing huge sums using Welsh geography. Lastly, the hosts encourage ongoing listener engagement, especially for resolving household mathematical quarrels.

Transcript

[Tim Harford]

You're listening to a More or Less podcast from the BBC. For more information about the programme, please go to the website BBC.co.uk radio4. Hello and welcome to More or Less, your weekly guide to the numbers in the news and in life. This week, we intervene in another mathematical marital misunderstanding.

[Julie (Listener)]

I've had an ongoing argument with my husband, who insists the odds of getting six consecutive numbers in the lottery are, ie, 1, 2, 3, 4, 5, 6, are higher than getting six random numbers.

[Tim Harford]

As the price of a train ticket rises yet again, we'll ask how the cost of UK rail travel really stacks up against continental opposition. But first, the conviction this week of Gary Dobson and David Norris for the murder of Stephen Lawrence 18 years ago is a good example of how important scientific evidence has become in the courtroom. The verdict hinged on tiny bits of circumstantial evidence, including clothing fibres and blood stains. One way to make sense of evidence like this is by analysing it statistically. How, for example, might Stephen Lawrence's DNA have got onto the suspect's clothes? What was the chance that contamination might have happened, as well as the small odds of someone else having the same DNA profile? After a while, though, these numbers can get confusing. Late last year, before the verdict in this trial was announced, I spoke to Angela Saini, a science journalist who's been investigating statistical mistakes in court. I began by asking her what sort of statistics tended to trip up the legal eagles.

[Angela Saini]

Well, when lawyers and forensic experts start applying odds to evidence, it's so easy to get things wrong that they even have names for the different types of errors. One's called the defendant's fallacy, for instance, because it's favors the defence. You may remember the trial of O.J. simpson in the US in 1995 for the alleged murder of his ex wife. So O.J. simpson had pleaded no contest to a charge of domestic violence, and this could have made him look pretty guilty. But his defence team claimed that this was irrelevant because fewer than one in a thousand women who are abused by their male partners end up being killed by them. Now, if you look at the stats from a different direction, you'll find that if a woman is abused and later murdered, there's actually an 80% chance that her partner did it.

[Tim Harford]

Okay, so these statistics, they can be quite powerful in a courtroom and quite deceptive as well.

[Angela Saini]

Exactly. So the most common error that you see is the prosecutor's fallacy, one that usually favours the prosecution. It's when the odds associated with a piece of evidence become confused with the odds of the suspect's innocence.

[Tim Harford]

Okay, my head's spinning.

[Ruth Alexander (Producer)]

So what does that mean?

[Angela Saini]

Well, let me give you a famous example from 1991, when a man called Andrew Dean was convicted for a rape in Manchester after he seemed to be a positive DNA match. Now, the DNA evidence was pretty vital in this case because, well, there just wasn't that much other evidence. Andrew Dean had no apparent connection to the victim. He even had an alibi from his girlfriend. But the jury was told that only around 1 in 3 million people would be a DNA match, and that made the DNA evidence look pretty compelling. The jury assumed that the chance of Andrew Dean being innocent must also be 1 in 3 million. So they found him guilty. But for a crime like this, the pool of suspects could be enormous, bringing down the odds of Andrew Dean's guilt quite dramatically. In a population of 20 million adult men, just for the sake of example, you'd expect there to be around seven matches. And this is why, in the end, Andrew Dean's conviction was quashed.

[Tim Harford]

So one way of avoiding this kind of fallacy would presumably be to analyse the statistics a bit more carefully.

[Angela Saini]

Well, that's right. The method often used is Bayes Theorem. In a courtroom, it forces you to logically examine how to revise your views of whether a suspect is more or less likely to be guilty given each new piece of evidence.

[Ruth Alexander (Producer)]

Okay, but if Bayes Theorem is the

[Tim Harford]

straightforward way of doing this, why don't courts use it?

[Angela Saini]

Well, there's the rub. Statisticians love numbers and they love Bayes, but judges don't. Courts have even ruled in the past that formally confused jurors. In fact, there was a murder case last year, R versus T, in which a judge went so far as to rule that Bayes Theorem shouldn't be used for certain evidence, including fibre matching.

[Tim Harford]

Angela Saini, author of Geek Nation but what exactly is it that the courts have objected to? I spoke to a friend of more or less Professor David Spiegelhalter of Cambridge University, and asked him to explain how Bayes Theorem actually works.

[Professor David Spiegelhalter]

Well, I must say, first of all, this is quite tricky stuff, and it's very easy to get it wrong. I think a Bayes Theorem is a formal way of using imperfect evidence to weigh up competing explanations. And those competing explanations might be, in a legal case, whether someone's guilty or innocent. And this is based on what's called the likelihood ratio, which is a fairly straightforward thing. It's the relative likelihood of the evidence under the different competing explanations. How likely is it to Observe what you did see, given all these different things that might explain what's going on. But the crucial element is not just this. You also have to take into account the plausibility of the explanations themselves based on other knowledge you've got.

[Ruth Alexander (Producer)]

So you have hypotheses coming in to Bayes Theorem before you introduce the new evidence. And the new evidence tells you how to update your views based on the new evidence.

[Professor David Spiegelhalter]

It tells you how to weigh the new evidence in order to change your opinions about these competing explanations.

[Ruth Alexander (Producer)]

Give us an example of how Bayes Theorem might have made a difference in a legal case.

[Professor David Spiegelhalter]

Well, the classic one is the Sally Clark case. A mother who had two of her children who sadly died and she was prosecuted for, for murdering the babies. And part of the evidence was how rare it was to get two cot deaths, two natural deaths in the same family, just by chance alone. Now, the expert very much overestimated the rarity of this, but that's, that's not quite the issue. The crucial thing is what is said is that it's the probability of this occurring if she really were innocent is really quite small. However, as we've heard previously, this is easily turned by the prosecutor's fallacy into a belief that that means that the chance that she's innocent is also very small. It's very easy to make that change in reasoning, which is completely wrong. We have to, by Bayesian thinking, take into account the plausibility of the other hypotheses and we have to look at how likely it is, for example, that a mother would murder her two children. And it turns out that if anything, that's even less likely than having two natural cot deaths in the same family.

[Ruth Alexander (Producer)]

So two cot deaths are hugely unlikely in the same family and, well, fortunately they're rare and most families do not suffer two cot deaths. However, given that two babies in one family have died, the question is not was that likely, but which is the more likely explanation, murder or natural causes? And that's where Bayes theorem comes in.

[Professor David Spiegelhalter]

Yes, we have to look at the relative plausibility. Given all the evidence that's in front

[Ruth Alexander (Producer)]

of us, how do you feel about the moves in legal profession to restrict the use of Bayes Theorem?

[Professor David Spiegelhalter]

Well, I'm very concerned about this. This R versus T case essentially has banned from the Court of Appeal the use of Bayes theorem and likelihood ratios in any areas apart from DNA and as they said, possibly other areas where there's a firm statistical base. But I think this is based on a fundamental misunderstanding of how probabilities arise, they seem to think they're sort of completely objective facts about the world that everyone must be in complete agreement about. But a group of top forensic scientists and statisticians have written about this and complained about this case, and if I can quote from what they say, they say probabilities should be informed by data, knowledge and experience. All data collections are imperfect and incomplete, and it necessarily follows that different experts might legitimately assign different probabilities to the same set of observations. This is a great way of saying that probabilities are really constructed on the basis of our knowledge and experience and we can't expect them to be, you know, completely, you know, iron hard facts that everyone agrees with. And as the forensic scientists point out, if you throw out the use of probabilistic reasoning, when there's ever any question about what the probabilities are, you're left with phrases such as, oh, this could have come from and is consistent with, which is, you know, ripe for, you know, to be misleading.

[Ruth Alexander (Producer)]

So really, the Court of Appeal appeared to be ruling that if probabilities are fuzzy, and actually probabilities almost always are a little bit fuzzy, then you're not allowed to process those fuzzy probabilities in a, in a proper mathematical logical way. You have to process them through intuition. And our intuitions are extremely misleading.

[Professor David Spiegelhalter]

Absolutely. Probabilities always are fuzzy to some extent. There's always some judgment that goes into their construction. And it would be really a huge loss if we threw out the whole possibility of rational reasoning, of evidence in legal cases just because we can't agree on some numbers to three decimal places.

[Tim Harford]

Professor David Spiegelhalter, you're listening to More or Less with me, Tim Harford. A number of you have written this week about rail fares which went up on 2 January. It was reported that on average fares had gone up by 5.9%. But many of you were skeptical of how this average was calculated and wondered whether the train companies had in fact worked out an average increase of ticket types, not taking into consideration that if they increased fares on popular routes and froze them on unpopular routes, their revenue could increase by much more than 5.9%. Well, all very confusing. Charlotte MacDonald is here to shed light on this. What's the basic problem here, Charlotte?

[Charlotte MacDonald]

Well, imagine a situation where a train company runs just two routes, one with one passenger and one with nine passengers.

[Tim Harford]

Ok.

[Ruth Alexander (Producer)]

This isn't exactly the rush hour crush, is it?

[Charlotte MacDonald]

Well, it's just a numerical example, Tim. Let's say each passenger is charged, charged 10 pounds so the total revenue is 100 pounds.

[Tim Harford]

OK, now what if the operator increases the busy ticket price by 10% and freezes the ticket price on the quiet route? Now, arguably the average ticket price has only gone up by 5%. There were two tickets and one of them went up by 10%. One didn't change at all. And I would sneakily claim, if I was running a train operating company, that prices had only increased by 5%.

[Charlotte MacDonald]

Well, you're a sneaky kind of person, Tim. But a more honest way to calculate an average is to look at total revenue. In this case, revenue would have risen from £100 to £109. So an alternative figure for the average fare increase would be 9%, not 5%.

[Tim Harford]

OK, well, it sounds like our loyal listeners are right to ask questions. So where does this real world figure of a 5.9% increase come from?

[Charlotte MacDonald]

Well, the number comes from ATOC, the Association of Trained Operating Companies.

[Tim Harford]

And do they calculate it the sneaky way or the honest way?

[Charlotte MacDonald]

Well, we think they're being honest enough. First, the train companies take a weighted average depending on how popular their routes are, and then ATOC takes that average and weights it again depending on the size of the company. The basic point is if a train operator raises the price of a peak hour ticket on a busy route, that price rise will carry a lot of weight in calculating the overall price rise.

[Tim Harford]

Okay, so the train companies aren't trying to pull a fast one. Still 5.9%, it's quite a jump in fares. And there has been a rash of articles pointing out that British commuters have to pay a lot more for rail travel than those in Europe.

[Charlotte MacDonald]

Yes, many of these seem to be based on figures put out by a group called the Campaign for Better Transport. They took five commuter journeys of around 23 miles from suburbs into major cities, taking London and comparing it with Paris, Berlin, Madrid and Rome. They then compared the prices and found that an annual season ticket from Woking to central London cost about 3,270 pounds, which compares pretty badly to all the European examples which were all under £1,000. For instance, it cost £340 to travel from a station called Velaytri to Rome and £650 to travel into Madrid from Colleado Villiarba.

[Tim Harford]

Okay, you say the distances are about the same, but are these really comparable routes?

[Charlotte MacDonald]

Well, that's just it. ATOC have pointed out that, for instance, if you're commuting between 7am and 9am There are 22 trains from Woking to London and the journey takes just 30 minutes, whereas the other four European routes took between 40 minutes and an hour to cover the same distance. And on the Italian route, There were only three trains between 7am and 9am, only nine on the German route, eight on the French one and 15 on the Spanish. So not as many as the lucky commuters from Woking can take.

[Tim Harford]

So this comparison is pretty arbitrary. And the routes are not particularly similar.

[Charlotte MacDonald]

No, apart from being a very similar length. This campaigning group are trying to make a point that fares in Europe seem cheaper than in the uk.

[Tim Harford]

And are they?

[Charlotte MacDonald]

Well, it depends who you are. I spoke to Roger Vickerman, economics professor at the University of Kent.

[Roger Vickerman]

I think it is difficult to say categorically that the British pay more. Clearly, in terms of the fare level, it is higher. But you've got to set this against the entire system, which is used for paying for the railways. Essentially, people will be paying higher taxes in a number of those other European countries to support the railway system. In Britain, we have decided it is the rail who should pay the higher portion of it. Also, of course, within most other European countries, fares have simply been set on the basis of the distance travelled, regardless of the time of day, regardless of the train that you're going on. Whereas, quite sensibly, in Britain, we've taken the view that we should try and price according to the level of demand. That means, of course, that commuters generally will pay rather more.

[Charlotte MacDonald]

In other words, you can land the bulk of the cost on rush hour commuters or you can ask other rail users and taxpayers to cross subsidize them. And in the uk, it's the commuters who tend to be asked to pay, which is bad luck if you're a commuter.

[Tim Harford]

Ok, thank you, Charlotte. In last week's programme, we served up a smorgasbord of data for our numbers of 2011. But we had this email via more or lessbc.co.uk, from Paul Clark, who was sure that our contributor, Matt Parker of Queen Mary University of London, had got something wrong.

[Paul Clark (Listener)]

I heard part of your Dateline straddling addition and caught the sound of a man airing the preposterous notion that the sum of all primes approaches infinity. In a strict mathematical context, that may be true. But in a program demystifying numbers that clearly can't be allowed to pass without comment, primes aren't the majority of all possible numbers. As primes become fewer, the higher their value, and while there may be an uncalculated number of them and their total may be Bogglingly large. It isn't remotely approaching infinity since a number nearer to it, almost certainly a lot nearer than the sum of all primes is the sum of all non primes. I think someone is confusing the concept of very big with infinite.

[Tim Harford]

Matt Parker sits before me in the dock.

[Matt Parker]

So, Matt, this is great because this is the wonderful thing about modern media. If you make a mistake, people will correct you very, very quickly. And this happens to me all the time. Although on this occasion, I think, strictly speaking, I was correct.

[Tim Harford]

Okay.

[Matt Parker]

Because what he said here is, if you think about it, the numbers are infinite. And that kind of makes sense because they keep going forever. And he says, well, not all the numbers are prime, and so there must be some, in some sense, fewer primes than numbers. And if that's smaller than the numbers, primes can't be infinite.

[Ruth Alexander (Producer)]

Yeah.

[Tim Harford]

So there's an infinite number of numbers. There's not an infinite number of prime numbers. And therefore, well, there are an infinite

[Matt Parker]

number of prime numbers. Okay, well, this is wonderful because even though Paul here is wrong, he's wrong in a very interesting fashion. And what he says matches the way that humans understand numbers, because we imagine numbers get bigger and bigger and bigger and bigger, and then eventually there's infinity as something a very long way away and very, very big. But in terms of maths, infinity is not just a very, very big number. Infinity is not an extreme. It's something that just never ends. And the normal natural counting numbers never end. They keep going forever. And the primes also never end. They keep going forever. In fact, you could count the primes, and there's the first prime, which is 2. The second prime is 3, and you can go all the way along. And in fact, the continue on forever in exactly the same way that the counting numbers continue on forever. They're both infinite, and they're both the same type of infinite. They both go on forever the same way.

[Ruth Alexander (Producer)]

Or I guess similarly, there are half

[Tim Harford]

as many even numbers as there are integers.

[Matt Parker]

Yes, but they're both infinite.

[Tim Harford]

They're both infinite. You could count them. Two is the first even number, four is the second even number, and it goes on forever.

[Matt Parker]

And you can count the square numbers. And in fact, it was in 1873, a guy called Cantor showed that you can count the rational numbers, which are all the fractions, and he found this lovely way of laying out the fractions in a nice kind of orderly queue and then going along and counting them one to the next one. He showed that, in fact, the rational numbers are the same Infinity as the countable number.

[Tim Harford]

So does this mean that all infinities are basically the same?

[Matt Parker]

There are different sized infinities and they can both be infinite because the countable numbers continue one after the other and they're infinite in that way. But if you look at, let's say, the real numbers, they also go on forever.

[Tim Harford]

So real numbers are basically anything, any ordinary numbers, including decimal points.

[Ruth Alexander (Producer)]

PI's a real number, decimal points, a real number, all kinds of irrational. They're all real numbers, they're all there.

[Matt Parker]

If you try to put those in a queue, if you had 2.5 and 2.6 in between those, you've got 2.56 and 2.57 and all these other little ones. And if you go between those, there are more. And it's subtly different to how there are a countable number of fractions, there's a bigger infinity of irrational numbers. They're both infinite, but they're infinite in slightly different ways.

[Tim Harford]

So there are two different kinds of infinity.

[Ruth Alexander (Producer)]

Well, this is where it gets even

[Matt Parker]

more interesting because the countables, we call that a left naught for the smallest infinity. And we know that's the smallest infinity, the reals. We're not sure if that's the next infinity up or if there's another infinity in between them. And we're still trying to work that out. It's an open question in maths in terms of how many infinities there are. We know that there are an infinite number of infinities.

[Tim Harford]

You say there are an infinite number of infinities. Are we talking about a small infinity or a big infinity?

[Matt Parker]

This is where it gets interesting again, because then you can argue which infinity are there an infinite number of infinities and then, well, actually there's an infinite number of different infinities. And then it gets slightly recursive at this point and it's absolutely amazing. It's all Cantor and Cantor sets and Cantor theory, if you want to look into it. And it's fascinating stuff, but the moral of all these infinite infinities is that strictly speaking I was correct.

[Tim Harford]

The stand up mathematician, Matt Parker. Well, I'm glad we sorted that one out, but it appears we made a very grave omission in last week's programme. We had this email again to more or lessbc.co.uk, from Howard Kaplan.

[Howard Kaplan (Listener)]

I'm one of your Canadian loyal listeners and today I heard the podcast of your December 30th show. In that show, Dr. Linda Yu discusses the projected Japanese public debt of 1, quadrillion yen. I'm disappointed that neither she nor you tried to express this large number in its most natural form as a fraction or multiple of the area of whales.

[Tim Harford]

A disappointment indeed. But thankfully, our loyal listener also filled in the gap that we had so carelessly neglected.

[Howard Kaplan (Listener)]

The 1 pound coin has a diameter of 22.5 millimeters, giving it an area of 0.000398 square meters. When circles are packed like hexagons, the packing density is 0.9069, so one can pack 2,281 coins per square meter. The area of Wales is 20,779 square kilometers, and if that area were covered entirely in 1 pound coins, the total value would be 47.4 trillion pounds. The current yen to pound exchange rate is 119.4. So 1 quadrillion yen is 8.37 trillion pounds, which would cover only 17.7% of the area of Wales with 1 pound coins.

[Tim Harford]

Well, thank you very much to Howard Kaplan of Canada, which, unless I'm very much mistaken, is about 480 times the size of Wales. Do please keep those questions and comments coming in time for a lottery draw, I think.

[Lottery Announcer]

Okay, and number one is our first number tonight of six balls in all, which is going to be next. In second place tonight, incredibly, it's the number two. And coming up next. Wow, this is quite something. It's the number three, would you believe? Never seen anything like this before. This is just fantastic. Number four is our fourth ball and. Wow, this is incredible. Our fifth ball of the evening is number five.

[Julie (Listener)]

And.

[Lottery Announcer]

Wait, no, this is a full set. Ladies and gentlemen, our sixth ball tonight is number six. And those are tonight's lottery numbers. 1, 2, 3, 4, 5, 6. In that order.

[Paul Clark (Listener)]

That's.

[Tim Harford]

That special draw was in honour of listener Julie, who emailed more or lessbc.co.uk this week.

[Julie (Listener)]

I've had an ongoing argument with my husband, who insists the odds of getting six consecutive numbers in the lottery I.e. 1, 2, 3, 4, 5, 6, are higher than getting six random numbers. I think the odds are exactly the same as for any six numbers drawn. Despite me explaining probability to him, he simply won't accept I'm right. Please convince him I know what I'm talking about.

[Tim Harford]

Well, fair enough. Kevin McConaughey, professor of statistics at the Open University, has offered to referee this domestic disagreement.

[Professor Kevin McConaughey]

Well, I know who's right. I'm afraid it's the wife and not the husband. The point is that Jule is quite right. That no number is preferred in the lottery, that is when the balls fall out of the machine they have, and so on. It's not that any one of them is more likely to come out than any other one. And it's also not true that if a particular numbers come out, that affects the chances of all the other ones coming out afterwards. And that's essentially a property of how the lottery machines work and they're kind of checked and lots of people keep an eye on this. So that's pretty well exactly true, as far as anyone can tell. Now, under those conditions, it means that any set of six numbers is just as likely to come out as any other set of six numbers. And what's kind of different? Different, though, is the way that we see those numbers. And perhaps that's what's confusing the husband in this case, because the chance of

[Ruth Alexander (Producer)]

getting 1, 2, 3, 4, 5, 6 in the lottery is minuscule.

[Professor Kevin McConaughey]

It's very, very small. It's about 1 in 13 million. But the same goes for every other set of numbers as well.

[Ruth Alexander (Producer)]

When they pop up. If it's, you know, 3, 9, 17, 21, 23, 30, we don't think to ourselves, that's just incredible. The chances of that, astronomical, although they were, the chances that that particular sequence came up were incredibly small.

[Tim Harford]

Doesn't seem remarkable.

[Professor Kevin McConaughey]

It doesn't seem remarkable because the numbers don't look particularly special. They kind of, in that the example you just gave, they look random, whereas 1, 2, 3, 4, 5, 6 or 7, 8, 9, 10, 11, 12, 13 or something like that kind of don't look random and you kind of think there must be something special about this, there's a kind of random process. So how can it give up sets of numbers that don't look random? Well, the fact is that it can and it kind of sometimes does.

[Ruth Alexander (Producer)]

And does that mean that if I was trying to win money on the lottery, which I have to say I would not recommend to any loyal listener as a money making strategy, but if I was trying to make money on the lottery, does it imply that it just doesn't matter what numbers I pick?

[Professor Kevin McConaughey]

The lottery itself is perfectly random and all the sequences are equally likely and so on, but people are involved and people are allowed to choose their numbers on the UK lottery, of course, and people are more likely to choose certain combinations of numbers than others. There's actually been quite a lot of research on what goes on now, a

[Tim Harford]

lot of this, actually, and I want

[Ruth Alexander (Producer)]

to avoid the combinations that other people have chosen because that means I would have to Share the jackpot.

[Professor Kevin McConaughey]

Yeah, that's right. Because choosing other people's numbers, numbers that other people have chosen, doesn't affect your chances of winning, but it affects the amount you win.

[Tim Harford]

Okay, so what numbers should I avoid then?

[Professor Kevin McConaughey]

What you've got to avoid is sets of numbers that form clear patterns. So avoid 1, 2, 3, 4, 5 and 6. Because roughly about 10,000 people, this is known in the UK, put that every week. Whereas there are plenty of combinations that nobody at all puts. I mean, you can tell, of course, when there's a rollover, nobody at all has put that combination. In many lotteries, the most popular choice, or kind of up there where the most, several, most popular, is 7, 14, 21, 28, 35, 42. That's going up in sevens. And people put patterns that kind of look nice on the lottery slip. You know, they kind of go down the middle or they go down the edge or something like that. But if you choose something that kind of looks terribly random, or ideally perhaps use the system that allows the lottery to choose the numbers for you, then that might be a better bet.

[Ruth Alexander (Producer)]

Kevin, I just want to take a step back for a moment. We have twice received emails from husbands and wives arguing with each other.

[Tim Harford]

Both times, I have to say, they've been fairly straightforward probability problems.

[Ruth Alexander (Producer)]

We've not been applying Bayes theorem, we've not been correcting for heteroskedasticity or regression to the mean. They've been pretty straightforward schoolbook problems. And both times the wife has been right and the husband has been wrong. Is this, do you think, statistically significant?

[Professor Kevin McConaughey]

Well, I don't think it is at all. You've only got two times here. And if we kind of assume that it's all random and it's, it's, it's, you know, it's perfectly random whether it's going to be the husband right or the wife right. In such and such a case, then we got an event that's happened that's, there's, there's a probability of half on the first one, there's probably half on the second one that the wife won. And assuming that there's nothing else going on, that's a probability of a half times a half or a quarter that it was a wife right. In both cases. And a quarter is not terribly unlikely really. So it doesn't really convince me of anything. But I would say that being a husband rather than a wife.

[Ruth Alexander (Producer)]

Well, our producer, Ruth Alexander, thinks it's, it's definitely a suspicious looking pattern, but I guess the only way to test this is to gather more data. So any loyal listeners out there, husbands arguing with their wives, wives arguing with their husbands, write in. We will adjudicate and we will keep track. Professor Kevin McConaughey of the Open University,

[Tim Harford]

thank you very much.

[Professor Kevin McConaughey]

You're welcome.

[Tim Harford]

And that's all we have time for this week, but please keep those emails rolling in with your questions and your comments and, of course, your marital disputes. We're at More or [e-mail] the website is BBC.co.uk more or less. And we're expecting to add some bonus podcasts this year. So if you're not one of the pod people, it's not too late to change that. In any case, there's one more program in this series. So until next week, goodbye. More or Less was presented by me, Tim Harford of the Financial Times. The producer was Wesley Stevenson, the editor, Richard Varden. And for more information and our terms and conditions, please go to BBC Co UK Radio 4.