Adventures in card play: mistakes with maths and how to avoid them when thinking about Covid-19 and Ukraine

So far as I know, I have only been blocked by four people on Twitter. Two are political charlatans who did not take kindly to remarks I had made about them (I’ve never directly interacted with one of them, so he must monitor commentary about himself assiduously). One is a charming and brilliant if intense tech guru who did not take kindly to my views on how we should use our personal power and influence in relation to others — if you must know, I believe in being fair, even when we dislike what someone has done in the past. The fourth was someone I managed to upset in a discussion about playing card probabilities.

This last one has been gnawing at me for the last few weeks, and I’ve worked out why. It tells us a bit about how people approach problems, a bit about how people approach Twitter and a bit about other real world problems. We have lessons we can learn for Covid-19 and Ukraine as a result.

It all started innocently enough. @AlanBaxter asked: “Tell me a simple fact that simply blows your mind.” Someone mentioned that if you properly shuffle a pack of cards, the probability of the cards dealt ever being repeated in the same order in a later deal were astronomical (being blocked, I don’t have the exact wording).

This was all I needed to make one of my favourite points. “Spartan if of the day. Most people can’t shuffle cards well, so in practice hands will recur regularly.” I have firm views on the poor shuffling of cards. It’s in that same category of niche concerns I have, alongside the correct use of paperclips.

My point proved controversial. Don’t you know just how mind-bogglingly huge 52! is, I was hectored. Atoms in the universe were mentioned, I recall. The lifetime of the universe was mentioned.

Most bitingly, I was told “Way to confuse pedantry with wisdom and suck the fun out something really interesting, Alastair. I bet you’re a hoot a dinner parties.” While perhaps I am not one of life’s raconteurs, I do like to hope that I can be kept on the dinner party list. I have good table manners as well. And, as it happens, I think my point is altogether more interesting than the original point.

For there is quite a bit of evidence that hands do recur, at least in practical card-playing terms. We reasonably often get reports that cards have been dealt in full suits.

When I pointed this out, my critics in turn pointed out that while this was staggeringly unlikely, it was not anywhere near as staggeringly unlikely as the original claim about the exact card order. My critics were therefore unpersuaded. After a bit more back-and-forth, the original poster blocked me.

I understand the maths that my opponents were putting to me. Honestly, I really do. But I have to say that their inability to review their assumptions disappointed me. For my linked example should have shown them that this was very possibly not just a pure maths problem. Perhaps, as William Hartson said, these were all down to pranksters. It’s not necessary to assume that, however; as Peter Rowlett explains in that linked article, many cases may simply be down to poor shuffling (coupled with the way that games like whist and bridge stack the deck). And bear in mind that most hands are neither particularly memorable or newsworthy, so if these reports of hands dealt in full suits are taken in good faith there are likely to be many other unnoticed instances of recurring hands.

Now if there are mechanical reasons why the pure maths don’t apply to hands as a whole recurring, it seems to me wholly unsafe to assume that pure maths apply to the order in which cards are dealt. It seems to me entirely possible, indeed likely, that the mechanical fault that led to hands recurring would lead to the card order itself recurring.

This can still be approached as a mathematical problem. The Reverend Bayes lies peacefully in Bunhill Fields, but torments the rest of us with his brain-melting logic. As Bayesians would say, we need to update our prior assumptions in the light of new information. If we find that something apparently massively improbable has taken place repeatedly, we need to assign a probability to the idea that something else is going on that we can’t see.

Nassim Nicholas Taleb, who so far as I am aware has no firm views on paperclips, made a fortune writing books like Black Swan which draw heavily on such ideas. For now, I just want you to hold in your head the idea that things that theoretically should be unconnected might in fact be very much connected. The dice might be loaded.

With that in mind, let’s turn to a question which has been much debated recently — how best to measure the performance of countries in dealing with Covid-19? The consensus has settled on excess deaths. I am afraid I think the consensus needs careful reappraisal, because excess deaths conceal as much as they reveal. The good news is that performing an intellectual striptease is very revealing indeed.

Let’s start with the case for excess deaths. Bodies, unlike diagnoses, are hard to hide. A doctor may or may not put Covid-19 on the death certificate. The approach to that decision may vary between doctors and indeed between cultures. In some countries, diagnosis of Covid-19 may have been inaccurate for a good part of the pandemic. But a dead body is a dead body and we can count those without requiring doctors to make judgements. Indeed, most countries have been doing so reliably for decades, so we can easily work out when deaths have increased.

The problem is that this approach assumes that all countries started from the same pre-pandemic starting position. That isn’t true. Some countries already had better health outcomes than others. We would expect that to affect the excess death statistics.

Let’s illustrate this with an example. Imagine a country called Erehwon, where healthcare is of a high standard, both in treatment and in prevention. Average life expectancy is 90. Next door is Catastrophia, whose healthcare is abysmal. Average life expectancy is 60. When Covid-19 hit, Erewhon would have had far more older citizens potentially vulnerable to the disease. Excess deaths might be expected, all things being equal, to be higher in Erewhon during the Covid-19 pandemic than in Catastrophia: the higher prior running death rate in Catastrophia (giving a higher base against which to judge the excess deaths caused by Covid-19) coupled with the proportionately more vulnerable population in Erewhon (among whom we would expect more deaths from Covid-19) should both work in Catastrophia’s favour.

So excess deaths need to be considered with that in mind. Countries with identical excess death rates should be judged against their prior health outcomes.

But in fact if we compare life expectancy with excess mortality in the pandemic years, we find that by and large the countries with the highest life expectancy also seem to have tended to have done better at dealing Covid-19, at least when looking at excess deaths. Italy and Spain are the two countries with high life expectancies which follow the pattern that might have been expected. Otherwise, it seems that long-lived countries had healthier Covid-19 outcomes. Among developed world countries, Britain and the US, for example, have done worse than, for example, France, Germany, Sweden and Japan.

I need to update my priors. There must be something else going on. It seems that, contrary to my expectations, the advantages of having a culture of healthful longevity are more important for dealing with Covid-19 than having a younger population.

This is very interesting indeed — even more interesting than my point about bad shuffling of cards. Perhaps good primary healthcare make a big difference. Perhaps well-organised later life care provision improved the chances of the elderly. Perhaps healthy lifestyles are important (obesity is known to be a big risk factor for Covid-19 outcomes, for example). In any case, there seem to be important lessons to be learned here about how to improve health outcomes.

If we were unwary, we might think that the benefits of having a good approach to health were already fully factored in and indeed might act as a perceptual disadvantage when working out how a country might do when looking at excess deaths from the new threat of Covid-19. That seems to be untrue and we need to find out why. It seems to be the case that having a good healthcare system usually helped a country cope with a new threat. That’s a lesson worth absorbing when considering how to load the dice.

This may seem far removed from the war in Ukraine. It isn’t.

When the war in Ukraine started, many (including me) anticipated that Russia would quickly overrun Ukraine’s defences. That did not happen. The Russians were poorly prepared, poorly disciplined, poorly equipped and poorly led. They have been driven out of northern Ukraine by a highly motivated and unexpectedly effective Ukrainian army.

The Russians remain well-entrenched in southern Ukraine and are apparently getting ready to focus their attention on the Donbas. We keep being told by military strategists that this is a very different proposition from the battleground in northern Ukraine and we should not assume that just because the Russians failed in their assault on Kyiv that they will fail to make progress in the Donbas.

I’m no general and I don’t pretend to have any knowledge about military strategy or logistics. I do, however, have a rudimentary understanding about Bayes theorem and it seems to me that it has a role to play here too. We again need to update our prior assumptions about what is going on.

We have learned that the Ukrainian military is far more effective than we previously thought. The reasons why are unclear from the outside and may include: greater motivation; effective training by NATO forces; effective management of supply lines; and behind-the-scenes assistance we can’t see. It would be helpful to understand exactly why the Ukrainian military has outperformed expectations, but we don’t have that information so we’re going to have to make do with what we’ve got. We have no reason to assume that their abilities are going to dip in a new phase of the war.

We’ve learned that the Russian military is much less effective than we previously thought. The reasons for this are a bit more obvious (success always gets less explanation than failure): corruption; ill-discipline; poor supply lines; poor equipment; overreach. Over the last month and a half, there have been few signs that the Russian military has learned from its mistakes. It still seems to be relying on brute force of numbers and materiel. Since both are still taking a hammering and Russian resources are exhaustible and being exhausted at a dizzying rate, that looks a very dangerous approach.

What I’m trying to say is that for those of us who are not generals, our expectations should be set partly by what the experts tell us but partly with an appreciation that we know how the dice are loaded. It may be that the Ukrainians have a far higher wall to scale now, but with sharply updated prior assumptions, we should rate them much more likely to scale it.

So it seems to me that the Ukrainians have a reasonable chance now of something approaching complete victory in the conventional war against Russia in the coming weeks and months. Quite what Vladimir Putin would order in response to such a humiliation is another question. We may well be due to find out in the not too distant future.

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Alastair Meeks

Alastair Meeks

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