Posts Tagged ‘probability’

Let’s talk about not believing in God.

Atheists often frame their position as a simple lack of a belief; they don’t take the active, affirmative, assertive position that theists do, don’t make any direct claim, and simply don’t hold the positive position that “God exists”.

I’ve written before about why the extent to which some atheists take this feels like an unnecessary cop-out.

Atheists should totally be making positive claims. Part of the reason why many are reluctant to do so, is because of an implicit idea that “belief” is a binary thing, something you either have or you don’t.

Christians believe the claim that “God exists”, and atheists don’t. Some atheists might conversely believe the claim “God does not exist”, but many deny holding any such position, and define their own godlessness as a kind of belieflessness. It’s not that they don’t believe in anything – we often have to remind people of our strongly held convictions in favour of love, truth, beauty, cheesecake, and the basic goodness of humanity – but when it comes to God, we simply don’t buy it, and are otherwise keeping out of the argument.

I don’t think this holds up. I think that the usual ways we describe belief are necessarily short-hand for a more complex set of ideas, and that we can afford to start being clearer in our positive declarations.

As an analogue, let’s say I’ve flipped a coin but not yet revealed the result. Do you “believe” that it’s landed on heads?

Assuming you have no improbable insider knowledge about the coin or my tossing abilities (steady), you haven’t a clue which way it’s landed. So, I guess you “lack the belief” that it’s landed heads. And you lack the equivalent belief that it’s fallen on tails. It’s not that you disbelieve either option – they’re both possible, and wouldn’t be especially surprising.

Now let’s say I’ve rolled a fair six-sided die, and am also temporarily hiding the results. What beliefs do you have about the number that’s showing? Presumably you lack each individual belief in its landing on any given number – but it seems like this is true in a different way from the coin-toss. In that first case, if you’d arbitrarily picked one option to guess at, it would’ve been no big deal whether you’d been right or wrong. With the die, if you randomly picked the right one, you’d be a little more impressed. On seeing what number it landed on, you’ve now adopted one particular belief you formerly lacked, just like with the coin – and yet this feels like a bigger deal.

Let’s step it up again. I’ve got a lottery ticket here for last night’s £7,000,000 jackpot. It’s either a winner or a loser, but I’m not telling you any of the numbers on it. Clearly you’d expect some evidence if I wanted to convince you it’s a winning ticket. But do you simply “lack the belief” that I’ve won the lottery, just like you “lacked the belief” that my coin had landed on heads (or tails)? Or are you actually pretty sure I haven’t won?

I’d argue that you’re easily justified in believing I’ve not become a millionaire overnight. The evidence in favour of the idea is so slight, and the odds against it so great, that it seems like a hypothesis worth ignoring. (Even before you consider the odds that I’m lying about having a ticket in the first place. Which I am.)

Now, you might change your mind later, when I invite you round for tea and diamonds in my new gold house, but for now, you’re safe assuming that I haven’t won the lottery. It’s not dogmatic to work with that assumption; it doesn’t imply you’re unwilling to be persuaded by evidence. But come on, clearly I haven’t won the lottery. Frankly, you should be quite content telling me “James, you have not won the lottery”. We’d all understand what you meant. If you can’t make that positive assertion now, then I don’t know when declaring anything to be true is ever going to be possible.

It may seem as if it’s incompatible with acknowledging the possibility that you might be wrong – this possibility can be calculated precisely, after all. But the fact is, we don’t append the phrase “to a reasonably high degree of probability, barring the arrival of any further evidence” to the end of every other sentence we utter. When we have conversations with each other, there’s generally a subtext of “I am not absolutely and immutably 100% certain that this is the case, it is simply the most appropriate conclusion I am able to draw and it seems strongly likely, but I will be willing to reconsider if there’s a good reason why I should do so” to most of what we’re saying.

I don’t “believe” that any given flipped coin has landed on heads or tails. But I can put a probability of 50% on either outcome, which says something more useful than just “I lack belief in any direction”.

With a six-sided die, the probability is 1/6 each way. Is it fair to say “I believe it hasn’t landed on 6”, since I believe the odds are 5/6 against that outcome? Probably not, but I don’t think it matters. If you understand the numbers I’ve placed on each possible outcome, you understand what I believe.

I don’t believe an asteroid is going to crash into the Earth tomorrow and wipe out humanity. Further, I believe an asteroid will not crash into the Earth tomorrow and wipe out humanity. I believe this more strongly then any of the other examples so far. How strongly? It’s hard to put an exact number on it, but that doesn’t mean it doesn’t belong somewhere on the scale of increasingly improbable things. In this case, just saying “it’s not going to happen” is a useful short-hand way to get my point across, without going into a lengthy spiel about percentages and Bayesian priors. It gets the gist of my position across in a manner I think most of my audience will understand.

There is no God.

Does that mean I think God’s existence is less probable than, say, flipping a coin and getting ten heads in a row? Would I be more surprised to meet Jesus after I die than to roll a string of double-sixes throughout an entire game of Monopoly? Whether or not I have an exact numerical value for my god-belief, these are the terms we should be thinking in. Not that there’s simply a thing called belief which some people possess and I lack and that’s the end of it.

So can we agree that a flat denial of God’s existence is not dogmatic and unfounded now, please? Can we accept all the implied background understanding that goes along with other conversations about the things we believe? Can we maintain useful phrases like “does not exist” without burying them under a mound of tentative qualifications each and every time, when we all know damn well that Carl Sagan’s garage is a dragon-free zone?

And could we stop acting as if being sure you’re right about certain things makes you an inflexible ideological bad guy, regardless of how reasonable it is to be strongly convinced of your position?

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Ben Goldacre’s got a fab example of misleading statistics, and the ways in which you can learn to think about things to avoid jumping to a wrong conclusion.

Look at his first nerdy table of data on that article. All they’ve done is take a bunch of people who drink alcohol, and a bunch who don’t, and counted how many from each group ended up with lung cancer. It turns out that the drinkers are more likely to get lung cancer than the non-drinkers.

The obvious conclusion – and (spoiler alert) the wrong one – is that drinking alcohol somehow puts you at greater risk of developing lung cancer. You might conclude, from that table, that if you currently drink alcohol, you can reduce your risk of developing cancer by no longer drinking alcohol, thus moving yourself to the safer “non-drinkers” group.

This is actually a fine example of the Bad Science mantra, and Ben makes an important point which many non-nerds might not naturally appreciate about statistics: the need to control for other variables.

If drinking doesn’t give you cancer, then why do drinkers get more cancer? The other two tables offer a beautiful explanation. Of all the drinkers and non-drinkers originally counted, try asking them another question: whether or not they smoke cigarettes. What you get when you do that is the next two tables.

If you just look at the smokers, then the chances of a drinker and a non-drinker getting lung cancer are almost exactly the same. If you look only at the non-drinkers, ditto. In other words, once you know whether someone smokes cigarettes, whether or not they drink makes no difference to their odds of getting lung cancer.

Which is a long way away from the obvious conclusion we were tempted to draw from the first set of data.

What we did here was to control for another variable – namely smoking – before drawing sweeping conclusions from the data. When we give smokers and non-smokers their own separate tables, it means that smoking cigarettes isn’t unfairly weighing the data we’ve already got any more. It becomes clear that drinkers aren’t simply more likely to get cancer; they’re more likely to be smokers.

And although Ben’s right to point out the importance of controlling for other variables like this, what interests me is the reminder of the importance of Bayesian probability.

In particular, the thing to remember is that the probability of an event is a measure of your uncertainty, and not something inherent in the event itself.

For instance, if that first table is all the data you have, then all you know is that drinkers are more at risk of cancer than non-drinkers. If you were to estimate somebody’s odds of getting lung cancer, and the only thing you knew about them is that they’re a drinker, the best you could do is to place it at 16% – the amount of drinkers who developed lung cancer in the study.

If you later acquire the extra data in the second tables, and find out that the individual you’re interested in is not a smoker, then suddenly you can re-adjust your estimate, and give them about a 3% chance of getting lung cancer. They haven’t done anything differently; nothing about their situation has changed for them to suddenly appear much more healthy. You’ve just learned more about them.

And it’s still not true that their odds of developing cancer are exactly 3% in any objective sense. Maybe tomorrow you’ll learn something about their age, or gender, or family history, and adjust your estimate again based on the new data. Maybe you don’t know that a doctor actually diagnosed them with lung cancer yesterday. This, obviously, makes a huge difference to their odds of having lung cancer – but it doesn’t change the fact that they’re in a low-risk group, and a 3% estimate is the best you can do based on your current knowledge.

In conclusion: stats are hard, listen to maths geeks (or become one yourself) before panicking about the latest tabloid healthscare.

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Don’t run away.

This post is going to be about maths and probabilit


There was a scientific paper recently published, in a respected academic journal, which purported to demonstrate evidence of human precognition.

Yep, science says people can tell the future.

Except, not really. Not yet, anyway. As the study’s author, psychology research Daryl Bem, said himself in the published paper, it was important for other scientists to repeat the experiment, and see if they got the same results. Richard Wiseman has been among those involved in such attempted replications, which so far have failed to support Bem’s original conclusion.

There’s a big moan I’m not quite in the mood to make, about how science generally gets publicised in the media, and the tabloids’ tendency to make a massive fuss over preliminary results, without concerning themselves with facts which later emerge and completely undermine their sensationalist headlines.

But I want to talk about the maths.

Replication is always important in science, particularly where the results look unlikely, or demonstrate something completely new. This is partly because, for all we know, Bem’s original research could have been dishonest or deeply flawed. Most people seem to consider both of these unlikely, though, and I’m certainly not suggesting that he’s faked his results.

But people often seem to assume that these are the only two options: that positive results must mean either an important and revolutionary breakthrough, or very bad science. The idea that something could just happen “by chance” now and then never seems to get much credibility.

Almost every time someone in a TV show or a movie proclaims something to be “just a coincidence”, or that there’s a “perfectly rational explanation”, we’re meant to take it as an ultra-rationalist denial of the obvious – usually supernatural – facts. Remarkable coincidences just don’t happen in the way that ghosts and werewolves obviously do. In fictional drama, there are good reasons for this. In the real world, this is a severe misunderstanding of probability.

When deciding whether or not to get excited about a result, scientists often look for significance “at the 5% level”. Bem’s results, supporting his precognition hypothesis, were significant at this level. But this does not mean, as you might think, that there’s only a 5% chance of the hypothesis being wrong.

What it means is: there would be a 5% chance of getting results this good, just by chance, if people aren’t really psychic.

So, getting results like this – statistically significant at the 5% level – is actually slightly less impressive than rolling a double-six. (If you have two regular six-sided dice, the odds of both landing on 6 on a single roll is 1 in 36, which is slightly less than 3%.)

I’ve rolled plenty of double-sixes. If you’ve rolled a lot of dice, so have you. And if you do a lot of science, you’d expect just as many random chance results to look significant.

So, if you’re thinking that we should probably ask for something a bit more conclusive than a double-six roll before accepting hitherto unconfirmed magic powers, you’re probably right.

This is the essence of Bayesian probability. Imagine having one of the following two conversations with a friend who has two dice:
“These are loaded dice, weighted to always land on a double-six. Watch.”

“Huh, so they are. Neat.”
“I’m going to use my psychic powers to make these dice land on double-six. Watch.”

“…Okay, that’s a little spooky, but you could’ve just got lucky. Do it again.”
You see why you might not believe it right away when your friend claims something really outlandish? But when it was something pretty normal, you’d be more likely to buy it?

In either case, the odds of rolling sixes by chance were exactly the same, 1 in 36, independent of what was allegedly influencing the outcome. But that doesn’t mean you should be equally convinced in either case when the same result comes up.

Both claims become more likely when the double-six is thrown. After all, if the dice really are loaded (or psychically influenced), then what you’ve just seen is exactly what you’d expect to see. But they’re not both getting more likely from the same starting point. One started out as a much more plausible claim than the other, and it’s still more plausible now.

Loaded dice? Sure, they have those. Telekinesis? Well, you have my attention, but let’s see you do it again. And again. And a dozen more times with a fresh set of dice.

This is part of my recurring, occasional project to convince the world that Bayesian probability is both important and intuitive, when it’s expressed right.

Ben Goldacre wrote about Bem’s research, the New Scientist also discussed it, there are some details of the replication attempts at The Psychologist, and I was prodded into thinking about all this in some more depth by a recent episode of the Righteous Indignation podcast.

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Yay, another maths lecture!

Click through to see the whole cartoon at XKCD. Really do it. It’s important. Especially if you want the rest of my burblings to make sense.

So. It’s partly funny because it satirises the sensationalism of tabloid news, and the urge to cram as much excitement into a headline as possible only to leave a sober assessment of actual facts to the blogosphere. But it actually addresses a much more common problem with our understanding of probability.

Most people who pay much attention to any kind of sciencey talk are probably familiar with the p-values referenced in the comic. When scientists are testing a hypothesis, they’ll often check whether the p-value (p for probability) of the results from their experiments is less than 5%. The smaller the p-value is, the less likely it is that their results are purely down to chance.

However, the p-value kinda means the exact reverse of what a lot of people assume it means.

When scientists talk about results being “significant at the 5% level”, say, it sounds like this means there’s a 95% chance of a real connection. In this cartoon’s case, it sounds like the scientists are 95% certain of a link between green jelly beans and acne.

Applicants for James Randi’s million dollar challenge are required to meet rather more stringent criteria, but it’s often expressed the same way. For instance, a dowser might have to psychically deduce which of several sealed containers is the one with water in, and repeat it a number of times, so that the p-value becomes very small. They want to be certain there’s really something going on, and it’s not just chance, before the money will be handed over.

But the intuitive idea of what the p-value means in these cases isn’t quite right.

Here’s what you actually need to do. Assume that there is no connection between the things being tested – jelly beans don’t affect acne, and all psychics are just guessing. Then, what are the odds of getting results at least as persuasive as the ones you saw, purely by chance?

That’s your p-value.

So, a p-value of 5% tells us something useful. It means that the results you’ve got are kinda iffy, given what you’d usually expect, if there’s no deeper underlying pattern there. You’d only expect to see results this skewed about 1 time in 20, if you’re relying on randomness. So maybe something’s up.

But if you do a whole bunch of tests, like the jelly bean scientists did, once in a while you will get some iffy results like that just by chance.

Now, clearly one thing this tells us is to be wary of data which has been cherry-picked, like the jelly bean journalists did. There were lots of negative results being ignored, and a single positive outcome highlighted. But the implications for how we assess probabilities more generally are, I think, more interesting.

In particular, it tells us that how likely something is doesn’t just depend on this one set of results. If a 5% p-value means “we’re 95% sure of this”, then this one study has entirely determined your estimate of the likelihood. It fails to take on board any information about how likely or unlikely something seemed before you started – and often this information is really important.

For instance, say you were studying differences between smokers and non-smokers, and the rate at which they get cancer. Any good analysis of data along these lines should easily pass a 5% significance test. It’s a highly plausible link, given what we already know, and 95% sounds like a significant under-estimate of the likelihood of a correlation between smoking and cancer.

But now imagine you’ve done a different test. This time, you just put a bunch of people into two groups, with no information about whether they smoke, or anything else about them, and flipped a coin to decide which group each person would go into. And imagine you get the same, seemingly convincing results as the smoking study.

Are you now 95% convinced that your coin-tossing is either diagnosing or causing cancer in people you’ve never met?

I hope you’re not. I hope you’d check your methodology, look for sources of bias or other things that might have crept in and somehow screwed up your data, and ultimately put it down to a bizarre fluke.

And it makes sense to do that, in this case, even despite the data. The idea that you could accurately sort people by cancer risk simply by flipping a coin is utterly ridiculous. We’d give it virtually zero probability to begin with. The results of your study would nudge that estimate up a little, but not much. Random fluke is still far more likely. If multiple sources kept repeating the experiment and getting the same persuasive results, over and over… then maybe, eventually, the odds would shift so far that your magic coin actually became believable. But they probably won’t.

And this idea of shifting the probability of something, rather than fixing it firmly based on a single outcome, is at the heart of Bayesian probability.

This is something the great Eliezer Yudkowsky is passionate about, and I’m totally with him. That link’s worth a read, though someday I’d like to try and write a similar, even more gently accessible explanation of these ideas for the mathematically un-inclined. He does a great job, but the arithmetic starts to get a bit overwhelming at times.

And if the thrill of counter-intuitive mathematics isn’t enough to convince you that this is fascinating and important stuff, read this. And then this.

Short version: a number of women have been convicted and jailed for murdering their children, then later released when somebody actually did some better statistics.

The expert witness for the prosecution in these trials estimated that the odds of two children in the same family both dying of cot death was 1 in 73,000,000. General population data puts the overall rate of cot deaths at around 1 in 8,500, so multiplying the 8,500s together gives the 1 in 73,000,000 figure for the chance of it happening twice. This was presented as the probability that the children could have died by accident, and thus it was assumed to be overwhelmingly likely that they were in fact deliberately killed.

But, as we learned with the cancer stuff earlier, we should consider these substantial odds against our prior assessment of how likely it is that these women would murder their children. This should start off minuscule, because very few women do murder their children. The fact that both their children died should make us adjust our likelihood estimate up a way – someone with two dead children is a more likely candidate for a child murderer than someone whose offspring are alive and well, after all – but it’s still far from conclusive.

Another way of expressing the central point of Bayesian probability is to consider the probability of A given B, for two events A and B. In this case, the odds of two children randomly picked from the population both dying of cot death may well be around 1 in 73,000,000 – but given that the children you’re considering both died in infancy, and were both siblings and so might have genetic or environmental factors in common, the cot death scenario becomes far more likely.

I wanted to expand on that last point some more, and touch on some other interesting things, but I’m hungry and you’re bored.

Ha. I said “briefly”. Classic.

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Scott Adams, the cartoonist behind Dilbert, posted a thing yesterday.

He was considering a step-by-step argument, which seems to result in the likely conclusion that life on Earth was the result of a deliberate seeding operation by aliens. Read it through on his blog before deciding it’s nonsense. I’ve summarised it very coarsely, and it’s more lucidly reasoned out than you might think.

His point, though, was to ask his readers to spot the flaw in the logic, which he finds himself unable to do, despite assuming apparently a priori that there definitely is a flaw. He doesn’t lay out explicitly why he’s unconvinced by what seems to him like watertight reasoning, and you may in fact be in agreement with the conclusion yourself.

But, a few problems with it did occur to me as I was reading, so I thought I’d try fleshing them out here, in a purely speculative and thoroughly uninformed manner.

– Firstly, I think the principle of indifference may be being inappropriately applied.

This is a mathsy thing. The idea is, you can basically guess equally between a number of possibilities when you don’t know anything about what’s going on, and simply have a number of options presented to you. If I ask you to guess what playing card I just randomly picked out of a deck, for instance, you might just as well say the nine of diamonds as the seven of spades. Nothing stands out about any one option, so you can apply the principle of indifference, and treat them all as being equally likely.

But sometimes it’s inappropriately applied. One way I’ve seen this done before is to argue that our Universe is likely to be only a simulation. We think we live in a reality that really exists, but as we approach a time when it’s feasible to create a Matrix-like simulation in which conscious beings could live unawares, we have to consider that maybe we already exist in such a simulation.

But maybe the reality that’s simulating us is itself only a simulation, within a reality which is also only a simulation, and so on, Inception-style, with as many layers as you like. Then, the possibility that ours is the real reality, and we just haven’t created any universe simulations ourselves yet, is just one among indefinitely many. So (the fallacious argument goes) the odds on that being the case are vanishingly small.

The reason it’s not convincing is that all the various options – that our reality is real, or that we’re the first simulation, or the second, or the seventy-fourth – should not be treated as equally likely. The idea that our reality is real makes fewer assumptions about the plausibility or the existence of colossal universe-simulating machines, and can legitimately be given a greater weight than the other options.

Scott’s argument may suffer from the same false application of this principle. It says: we could soon be the first species ever to send spaceships to other planets and “seed” them with the building blocks of Earth-like life – or we could be one of many stops in an indefinitely long chain of other species which have already done that. That is, Earth may have been seeded by an alien civilisation, which itself was seeded by another, and another, and so on.

If you consider that we could be at any point in the chain, and treat them all with the principle of indifference, then it may seem unlikely that we just happen to be the first, “unseeded” life-forms in the cosmos. But there are different assumptions involved in “It’s already happened” than “It hasn’t happened yet”, and so, barring any other evidence which directly supports it, I don’t think we’re obliged to give the possibility that our own world was “seeded” so much weight.

– Because, don’t forget, there is no other evidence directly supporting the idea that this seeding is what’s happened here. However solid the arguments might be that it could happen, or that it’s virtually inevitable to happen with any life that reaches a certain threshold of intelligence, it’s all just speculation. Nothing wrong with that, but it’s not the same thing as empirical data. However unlikely you want to argue that the “unseeded Earth” possibility is, it’s entirely consistent with the current data, and it makes fewer assumptions about the Universe than the alternatives.

– I’m also not fully convinced that any intelligent life-forms would necessarily reach the point where this seeding of other worlds becomes both practical and desirable. There are various assumptions on which this rests, like our (or other life-forms’) ability to get that far technologically without destroying ourselves; the superior plausibility of the seeding option over any other methods for sustaining life; the eventual success of even a well planned seeding mission in giving rise to intelligent life again; and the timescale necessary for this to happen. (We have pretty good evidence that life on Earth has been evolving slowly for about a quarter of the age of the Universe. It can’t have happened that many times, going by this iteration rate.)

– We also have no idea how likely the possibility of alien life actually is. There’s so much uncertainty over so many variables of the Drake equation, that whether or not any other life has yet been able to arise anywhere else in the galaxy is still deeply contentious. A lot of things needed to be exactly right on Earth for life to get going and start becoming complex and interesting, and we don’t really know how rare those conditions are. The scenario of other aliens having got there before us is far from being a given.

Leave a comment if there are any more obvious points leaping out at you which demonstrate that one of us is going wrong.

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Quick, think of a colour and I’ll try and guess what you picked.

Red. No, yellow. Orange. Yellowy-orange. Wait, green. Definitely one of those greenish-blue, blueish-purple kind of colours. Like blue. Purple? Did I get it yet?

Okay, now think of a number between one and ten.

Got it?

Seven. No, five. Okay, but it’s definitely odd. Definitely… I’m getting strong vibes that it’s even, like eight, or six… or maybe three. Two. Am I close?

Okay, it’s not that impressive, but if you ignore all the times I was wrong, I think you’ll find that I did really quite well.

And this is something that people are often very good at doing, if they’re not really putting together a thorough investigation – if they think they know what’s right, and are just looking for further evidence to prove their ideas correct. Although I doubt the example above provoked anybody to leap to their feet and shout “By all the gods and underpants gnomes, he’s right, I was thinking of blueish-purple!”, it can be surprisingly easy to overlook and forget about large amounts of data, even under less obvious circumstances.

You might have a story about a time when you got an unexpected phone call from somebody, out of the blue, right when you happened to be thinking about them. It’s the sort of thing that can seem like an impossible coincidence, especially if it’s someone you haven’t heard from in ages, who you had no reason to be thinking about, and who you never would have expected to actually call you. You thought about them, then suddenly there they were – seems like pretty strong evidence in favour of something.

But fleeting memories of quite a number of people, who may not play a major role in your life any longer, probably flit through your head each day – and you probably get spontaneous and unexpected phone calls or emails every so often, too. Once in a while, you’d expect these things to somewhat match up.

It shouldn’t be surprising when something spookily coincidental does occasionally happen, because the criteria for what’s “spooky” can be fairly wide. Does it only count if you get the call the very same minute you think of them? Or within ten minutes, or twenty, or an hour? If you dream about someone, does it still count if you hear from them any time in the following day, or week? This can be extended to any other allegedly prophetic dreams, as well. My subconscious comes up with some pretty freaky shit most nights, and every time the nightmare about being chased over a hill by the giant peanuts doesn’t turn out to have any bearing on reality, that’s a data point too. Those happen a lot more often than ones where I do seem to be able to predict the future.

Also, there are a lot of people for whom this sort of thing just doesn’t seem to happen, and that’s important data as well. I don’t recall ever having a dream which seemed to have any predictive power, even after the fact, or being surprised by any kind of communication from someone I was just thinking of. There are lots of people who do have impressive-sounding stories, but with more than six billion people on the planet, how rare do you think million-to-one chances really are?

Psychics and cold-reading are going to get a lot more coverage here in future articles, but remembering the hits and forgetting the misses is a big part of that whole routine. The second part of that video shows Michael Shermer laying out pretty clearly how much random guesswork is involved in a typical reading, and quite how readily some people will be willing to ignore the many wrong answers, as if they didn’t tell us anything, and only notice the hits that were eventually stumbled upon, as if they were a lot more impressive and inspired than they really were.

Astrology‘s full of this. All sorts of general stuff easily glossed over, and the occasional match with something we can relate to, which then stands out as if it were representative of the predictions as a whole.

Human beings are not natural statisticians. We’re often shown up as having a very poor innate grasp of probability, and very little intrinsic ability to reason things out. The birthday paradox is one of the clearest examples of this, and I still have trouble getting my head around the fact that it only takes twenty-three people together for it to be more likely than not that two of them have the same birthday. I know it’s an old one, and I totally get the maths which makes it clearly true, but it still messes with my head. There are probably over 23 people in the office where I’m working at the moment, but if any two of them happened to discover that they share a birthday, I’d still feel a strong instinct to go, “Ooh, neat,” as if this were much more unlikely than it is.

I’m good at maths, and I still suck at instinctively estimating this kind of thing. This is just another reason why analysing data closely is important if we’re going to draw any dramatic conclusions, and our natural initial reactions of how impressive something seems should not be held in any particular esteem. People can be easily impressed by all kinds of junk if they’re inclined only to count the hits; the misses are easily forgotten, but they matter too. Van Praagh is doing nothing astounding when you see all the data, and it’s only then that we can hope to make a fair assessment of whether anything remarkable is actually happening.

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This doesn’t directly relate to any particular skeptical topic, but it seems to come up somewhat obliquely in reference to many different ideas, so I thought it’d be good to gather a few thoughts into one place.

It’s a common mantra among some people that “There are no coincidences”. What this generally means is that, whenever something seems serendipitous, or appears to have come about by chance in a particularly orderly way, or contains an unlikely pattern reminiscent of something else, or is in any other way quirkily coincidental, then you can bet that there’s something deeper going on, something we’re not seeing but which has caused things to align themselves in such a pattern, something that deliberately made things this way. Anytime something seems oddly out of place, or circumstances are just a little too “convenient”, you should be suspicious, and try to find out what’s really going on under the surface. There are no coincidences.

My position, however, is that if there really were no coincidences, this would be the most phenomenal coincidence imaginable.

Well, think about it. Imagine there are no coincidences. Not a single one. If you ever get talking to someone at a party, and find out that you both have the same birthday, then the two of you must have been brought together for a reason. It’s impossible for anyone to ever just stumble across any one of the millions of other people who were also being born around the same time that they were (or even on the same day on a different year), simply by chance. It could only happen when some underlying force makes it happen.

Of all the thousands of fleeting thoughts that pass through a person’s head each day, nobody could ever find any possible correlation between the vague and unprompted recollection of a person or place they haven’t thought about much in a while, and a phone call or other physical reminder of that same person or place, unless the thought was deliberately put in your head by some force that knew you were about to get that phone call. It could never happen that your idle musings just happened to overlap with reality by chance.

Out of the entire human history of bits of fruit that have gone slightly mouldy, or bits of cheese that have got burnt, not a single splotch of scorched dairy or unripe tomato could ever possibly have naturally curved itself into a pattern that sorta kinda looks like a person’s face a bit. You only need to look at the ubiquity of emoticon smileys to realise how low the human brain’s threshold is for spotting another human face (more on pareidolia soon), but even so, any simple pattern with elements that seem to remind us of facial features, like a couple of dots for eyes and a bit of a curve for a mouth, could never ever have simply come about by a random process of creating splotchy patterns, like burning a bit of cheese or letting mould grow in a tomato. It must have been placed there for some deliberate purpose.

Nobody could ever stand in front of an audience of people who are all hopeful to hear some good news about their dead relatives, and just happen to guess that the name John might mean something to someone. If anyone in a crowd of several hundred people is called John, or knows someone called John, or ever did know someone called John, then there’s no way it could just be a coincidence that a psychic happened to pick that specific name. Clearly whenever someone has such knowledge, it can only be through true spiritual intuition and guidance.

However often millions of people check their bank balances, at no point will any of them ever discover that the number happens to be in some numerical sequence that stands out to us, like £666.66, or £1234.56, or any of the other uncountable ways we could find patterns in such a string of digits. It’s just not plausible that a massive sample of random numbers constantly fluctuating up and down in varying degrees could ever land on any of the possible results that look pleasing to us. It must be portentous of something.

Oh, and those Rorschach inkblot tests? No way is it a coincidence that a bunny with a chainsaw was staring right at me in three consecutive cards. Whatever I’m getting an image of amidst the random visual “noise” of the inkblot, someone must have put it there deliberately.

Coincidence can clearly take an extremely wide range of potential weirdness, and at the mundane end we’re unsurprised to bump into them all the time. I don’t freak out about cosmic synchronicity every time I meet someone who shares my first name; it’s a pretty common first name, so obviously it’s going to happen a fair bit. Less likely is that I’ll find someone who shares my birthday, but 1 in 365 (ish) still isn’t exactly suspicious, and you only have to put 23 people in a room together to make it more likely than not that some of them will share a birthday.

On the other hand, there can be some big coincidences, that really do seem too good to be true, too profound and improbable to be simply down to chance. Lightning is a perfectly natural phenomenon, but if a bolt of it came crashing down through my roof and vaporised me the very moment after I declared “I swear, I was only holding that porn for a friend, and may God strike me down where I stand if I’m lying,” then this might raise even the most skeptical of eyebrows. If such an unlikely coupling of events did coincide, then we might really be persuaded to look for a deeper cause, which could have brought things about more plausibly than plain luck.

But the very fact that you can see a difference between the nature of these two situations – the former of “Hi, I’m Steve,” “Hey, me too”; the latter an impeccably timed lightning bolt – clearly demonstrates that some kind of judgment call has to be made here. We all expect some level of freaky coincidence to just happen, and we have every reason to expect the random noise of any media to produce some unlikely-seeming patterns now and then. Look far enough into the digits of pi, and you might find your telephone number. Flip a coin often enough, and eventually you’ll get ten heads in a row. Nearly every week, someone in the country overcomes millions-to-one odds and wins the lottery. And if any of these is simply a silly example, and your personal preferred coincidence is much less frivolous, then you’re performing some sort of evaluation to determine the weirdness, and to assess just how implausible the “coincidence” explanation is.

If you find conspiracy in every purported coincidence – literally any time two or more artefacts “coincide” – then you’ll never have the time to notice anything else. What’s important is to have enough of a mathematical understanding to distinguish genuine weirdness – where random chance truly becomes less likely to have caused something than deliberate intent – from the times when slightly kooky stuff just happens in an entirely expected way. We all make those judgment calls. It’s just a matter of whether you’re sufficiently informed and equipped to make them well. Actually looking at the odds and figuring out how suspicious to be of something is always better than trusting your gut and going with what feels more likely. Just ask Monty Hall.

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I started writing this entry ages ago, back when the subject was vaguely topical again amongst a few of the ScienceBloggers, but because I still suck at blogging, it’s taken this long to get around to finishing it. But I wanted to put something coherent together about this, because although I’ve made several off-the-cuff attempts to explain it during various conversations before, I’ve never yet organised my own thoughts on the Monty Hall problem into an actual blog post. So, here goes. Fretter ye not: there will be as little actual maths here as possible, although there is a lot of head-bendy number-wrangling to get to grips with in the problem itself, and it is interesting whatever any of you say.

What’s the dealio?

Imagine that you’ve used all your wit and cunning to make it through to the final round of a game show, eliminating the competition mercilessly with your infallible knowledge of pointless trivia and/or ability to press a button really fast. The rewards for your victory so far are numerous and many, but pale in comparison to the grand prize… A very, very shiny thing. Oh, such glorious dreams that might finally be realised this night.

And the path to true delightenment is so very simple. There are three doors. Behind one of them is your shiny thing, garnished with an elegant silk ribbon and all ready to go. Behind each of the other two is a small tuna sandwich, as a rather sad and pathetic consolation prize. You’re not that hungry, but you do like shiny things, so it is vital that your final reserves of tactical brilliance are brought into play now.

Your first task in this deadly game is to pick one of the doors. This door will not be opened, but held provisionally as your first, tentative choice. You’re not committed to anything yet.

Next, I, your charismatic and photogenic host, will show you where one of the sandwiches is to be found, by opening up one of the other two doors. Your initially chosen door will not be opened, and the grand prize will not yet be revealed.

Then, you simply get to open one of the two doors that remain closed; whatever lies behind is yours to keep, whatever remains hidden behind the third door is lost to you forever.

Your choice, of course, is between resolutely and firmly sticking to your convictions and opening the door you chose in the first place, or flip-flopping indecisively and picking the other one. So, which is it to be?

Well, if you let my ever-so-subtly inflammatory language sway you, then bad luck. If you stick with your original choice, you’ve got a 1-in-3 chance of winning. You can double those odds, to a much more friendly 2-in-3 (I know I said there wouldn’t be much maths, but I really hope you’re not struggling already) if you switch your choice, and open the other door.

Curse you, logic!

So, what’s up with that? After that first sandwich is out in the open, it looks like a fairly simple case of 2 doors, 1 cup shiny thing1. Your prize is either behind one door or the other, so it’s just a 50/50 decision, right? It shouldn’t make any difference which of the two doors you pick, because you’ve got a 50% chance of picking the one with the prize in each case, right? So it doesn’t make any difference whether you switch or not… right?

No. Not right. The opposite of right. (There should be a word for that.) Hence the controversy. There’s some great stuff on the wikipedia page about the thousands of people who wrote to Marilyn vos Savant to tell her she was wrong, “including several hundred mathematics professors”, and a great deal of debate still rages on, as evidenced by the discussion page for that article. Some of this active disagreement is to do with important semantic details of how the game works, but a lot of it comes from people thinking they’ve managed to outsmart hordes of mathematicians by saying “but it’s obvious!!

So although you can embugger about with the precise wording and turn it into a different problem with a different solution, the way I’ve phrased it above is the best-known form, as well as probably the most counter-intuitive and perhaps the most interesting. So, let’s see if I can persuade you that you’re wrong in time for tea, when I will proceed to use the Banach–Tarski paradox to prove that my slice of cake is actually equivalent in size to yours, even though it appears to have twice the volume.

… or door number three?

The semantics are important here, and it’s worth thinking about where that “1-in-3” number even comes from. What is it describing? Obviously it relates to the probability of a shiny thing being behind a door, but it’s not strictly true to simply say “The probability of the prize being behind this door is 1-in-3”. Whichever door you happen to be pointing at, the odds of the prize being behind it, actually, are either 100%, or 0%.

Your prize isn’t hovering behind each door in some misty field of quantum indeterminacy like a half-dead cat in a box, and opening the doors doesn’t collapse any wave-functions to change the probabilities. If you open door number 2 and find a sandwich, the odds of that door hiding the prize don’t then suddenly become zero; you never had a hope with that door, let alone a 1-in-3 hope.

Understanding this might help you get around some of the more awkward sloppy thinking that gets in the way of understanding the paradox. What the 1-in-3 odds refer to is the chance of a randomly chosen door turning out to hide the prize. If you’re about to close your eyes and point, you’ll have a 1-in-3 chance of choosing the right one, because if you repeated the experiment numerous times, you’d get it right one time out of every three.

Similarly, when there’s two boxes left, picking one randomly would give you a 1-in-2 chance of winning. But you’re not doing that. You can do better than just make a random guess, because you have more information available to you, so you can improve your odds. I’ve cut a lot of wordy rambling from the first draft of this article at this point, because I think it’d be more helpful to dive in with a semi-practical example.

In a world of pure imagination

Close your eyes and picture a scene. Wait, don’t go drifting off into your own fantasy-land yet, I’ll describe it to you first. It’s okay, this isn’t an excuse for me to divert your attention while I creep up on you and throw a massive spider onto your face and laugh and laugh. It just helps to have a visual for this, and waving an actual deck of cards at you isn’t really an option from where I’m sitting. (After all, not only am I far away, but from my perspective as I type this, you’re reading it in the future. Spooky.)

Anyway, close your eyes if it helps, and imagine I’m waving a deck of cards in your face. We’re going to play a variation on the classic Monty Hall game, I tell you. (Don’t worry about how I got into your home with these cards in the first place. It’s all fine. Just relax. There is no massive spider.) In this variation, there are cards instead of doors, and fifty-two of them instead of three. The ace of spades represents the prize, the others are less rewarding than a Jo Caulfield stand-up show. (Not witty, hardly scathing, but I haven’t mentioned to anyone lately that Jo Caulfield isn’t funny, so it had to be done.)

Much like in the original, you start off by picking one of the cards. You pull it across the table (yes, there’s a table now, don’t strain your imagination) toward you, keeping it face down and unseen, while I hold the other fifty-one cards. I then look through my cards, and throw fifty of them face-up on the table, all of which are not the ace of spades.

So, do you stick with the one you first chose, or would you rather change your mind and have the one I’m left with from my batch of fifty-one?

Hopefully it’s obvious why I hope this answer is obvious. After you made your initial pick, and before I showed you those fifty cards there, it would have been pretty clear that I was shuffling through looking for the ace of spades, so that I could hold that one back and throw the rest of them down. It’s possible that you happened to pick the exact right card on your first try – your odds were 1 in 52, which isn’t really that remote at all – in which case, when I looked through my 51 cards, I would’ve seen that I could’ve chosen any batch of 50 to show you, and kept any one of them behind, and you’d have been better off sticking.

But it’s far more likely that you grabbed something other than the ace of spades the first time, leaving me with that ace along with fifty other cards, so I had no choice but to show you those fifty. If someone else was brought in to the game at this point, and just saw two cards face-down on a table and tried to pick the ace, they could do no better than to guess at random with 50/50 odds. But you know more than they do about these two cards: you know which one of them you picked at random, and which one I (probably) selected carefully. So you know that sticking with your first choice means that you’re relying on 1-in-52 odds, which you can deliberately choose to avoid, giving yourself a 51-in-52 chance of coming out on top.

Probability is weird. Our brains aren’t wired to be able to naturally handle complex and unnatural likelihood analysis. This is meant to be confusing. Even if you get that several reds in a row doesn’t mean that a roulette wheel is “due” a black on the next spin, this stuff is hard. Don’t sweat it.

1 One of the less successful spin-offs, currently with only 103 views on PornoTube.

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