Mar 25 2010

The Flake Equation

XKCD, a science-savvy webcomic, has a recent installment called The Flake Equation (a play on the Drake Equation which provides the formula for calculating the likely number of technological civilizations in the universe). I am now going to take his brief and elegant point and beat that horse to death.

While XKCD applies his equation to UFO sightings, it applies equally well to any paranormal or pseudoscientific phenomenon – bigfoot, ghosts, alternative medicine healings, etc. Often we are told by believers that “where there is smoke there is fire.” In other words – with so many people reporting sightings or healings, there must be something going on.

Skeptics, of course, recognize this as a fallacy. There is a major unstated premise or assumption in this position. We must ask ourselves – in a hypothetical world in which we are not and have never been visited by aliens, what would we expect in terms of sightings and experiences? Is such a world compatible with the current number and quality of sightings? I think the answer to this is a pretty clear “yes” – and that is exactly the point that XKCD is making.

There are a host of psychological, cultural, and situational factors that would lead people to believe they have had a paranormal or extraterrestrial encounter. Most of us have had weird experiences we may or may not have been able to explain.

We add to this the fact that human memory is both flawed and biased toward interesting and meaningful narratives. We tend to morph memories over time simply because they fade with time, and are susceptible to merging with other memories and even contamination. Have you ever “remembered” an event that you never experienced but someone else told you about. You imagined the story, and then years later confused your imaginings with your own first-hand memory.

Further, we tend to smooth over the rough edges of our memories – distorting details to fit the theme. So if we think that our encounter with a strange light was an encounter with an alien spacecraft, over time the details of the story will fit a UFO encounter better and better. We will drop or alter disconfirming details, and enhance or even confabulate supporting details. Details from other witnesses will also contaminate our memories (and may give the false impression of independent confirmation).

In the end we have a story that sounds very compelling, and the person telling the story likely really believes it and the supporting details. Often believers are compelled by the sincerity of those with such tales to tell. First, sincerity can be faked. But even if we assume it is genuine, it is meaningless because people can be sincerely mistaken.

XKCD makes the further point that given the number of people in the world we would expect a large number of even unlikely events (mundane but unusual or highly coincidental), and some of this will be further distorted in the telling, and some of these will become widely spread.

The take home message for skeptics is that even in a world without ghosts, aliens, and bigfeet we would expect there to be large numbers of anecdotes that sound very impressive, told by people who seem sincere. In a world with these things, we should also expect some objective and verifiable evidence – and that is what is lacking.

The notion of “where there is smoke there is fire” simply does not apply, because human brains and the cultures in which they are embedded are smoke machines.

33 responses so far

33 thoughts on “The Flake Equation”

  1. modoc451 says:

    I wonder what the number would be in the U.S. for the last variable.

    @Steve: Are you a regular reader of xkcd?

  2. Plittle says:

    I’m somewhat confused, Steve. Are you contending that the XKCD comic in question was putting forward the ‘where there’s smoke, there’s fire’ argument? I think you’ve read it wrong. Did you read the roll-over text?

  3. jblumenfeld says:


    I think you missed one of the major points of this cartoon – funny, because its a pretty direct indictment of the Drake equation itself. The massive assumptions of the Drake equation can get you pretty much any answer you like as to the likely number of technological civilizations, and it needs to have finger poked in its eye occasionally.

    Yet another reason to love XKCD.

  4. avar says:

    Plittle, jblumenfeld: You’ve misread Steve’s post. He says he’s going to “take his brief and elegant point and beat it to death”. To beat something to death has a repetitious connotation akin to “beating a dead horse”. He’s agreeing with the comic but presenting the point it’s making in a more general and verbose manner.

  5. BenjaminB says:

    Randall Munroe almost certainly wasn’t serious, but some of his readers may have taken him seriously–XKCD sometimes expresses genuine ideas, it’s not always jokes–and it’s good to remind people why equations like these need to be taken, at the very least, with a grain of salt.

  6. atomb says:

    As far as I can tell the Drake equations was set up to show what we would need to know to calculate the chances of life. It doesn’t need a “stick in the eye” only people who fill in the drake equation with junk need a poke. Plittle – I’m pretty sure Steve gets it and is taking the same viewpoint and expanding it to cover other magical stories.

    Thanks Steve, great article as usual!

  7. Suraky says:

    jblumenfeld … I think you would have to show intent to indict Drake for that crime. We all know the Drake equation is just a thought experiment.

  8. Plittle – no, I am saying the opposite. That XKCD was providing an argument against the “where there’s smoke there’s fire” argument.

    jblumenfeld – I disagree. The Drake equation itself does not offer any assumptions or even an answer – it just identifies the variables. You can argue with what figures to plug into those variables, or even if we can know at this point what they should be – but the equation itself is valid.

    I don’t think we can infer any criticism of the Drake equation from this cartoon. BTW – did you see the new South Park episode? By coincidence it deals with over-interpreting text according to our own assumptions.

  9. jblumenfeld says:

    Please. The Drake equation SHOULD be just a thought experiment, but from the very beginning numbers got plugged in and it got used as an argument in favor of a major SETI effort. I’m not saying SETI is a bad thing, but let’s not draw some kind of wall of purity around the Drake equation. It has been widely abused from the start, in exactly the way XKCD illustrated it could be abused for something less popular among scientists.

    The defensive reaction my post engendered is just proof of his point.

  10. Who’s being defensive? You are now making a habit of reading what you want into things.

    Your point is a non-sequitur. Abuse of the Drake equation does not invalidate the Drake equation. You should be careful to make that distinction, and limit your criticism to what you consider to be the inappropriate use of the Drake equation.

    Further, there is nothing in the XKCD cartoon to support your inference. He is, in fact, putting an alternate version of the equation to good use – as a back-of-the-envelope estimation. As he wrote – even if we plug in conservative numbers, we would expect there to be large numbers of compelling UFO reports even without aliens.

    If he intended to lampoon the Drake equation he should have used a version of it to come to an absurd conclusion, not a reasonable conclusion.

  11. jblumenfeld says:

    For example, here are the values that Drake proposed as conservative possibilities – in 1961, at the same time he proposed the equation:

    Considerable disagreement on the values of most of these parameters exists, but the values used by Drake and his colleagues in 1961 were:

    R* = 10/year (10 stars formed per year, on the average over the life of the galaxy)
    fp = 0.5 (half of all stars formed will have planets)
    ne = 2 (stars with planets will have 2 planets capable of developing life)
    fl = 1 (100% of these planets will develop life)
    fi = 0.01 (1% of which will be intelligent life)
    fc = 0.01 (1% of which will be able to communicate)
    L = 10,000 years (which will last 10,000 years).
    Drake’s values give N = 10 × 0.5 × 2 × 1 × 0.01 × 0.01 × 10,000 = 10.

    (from wikipedia)

  12. Again – the only problem is assuming more precision than we currently have. It is a tool of estimating assuming plausible values, but with recognized error bars. Yes the error bars are huge (especially cumulatively) – but as long as that is being admitted, there is no problem.

  13. jblumenfeld says:


    I’m making a habit of reading what I want into things? That’s not very nice. And speaking of that, I never said the equation was invalid. I said it was abused – in exactly the way XKCD is showing abuse of the ‘flake equation.’ You read that into my post. I did exactly what you said I should do – I talked only about the abuse of the equation. Quote me where I said it was invalid. I said the massive assumptions can get you any answer you want – you know I’m not the first person to say that.

    I also don’t think I’m being defensive, I’m pointing out that the Drake equation has been abused, and I think XKCD is perfectly capable of making that point without hanging a sign on the cartoon that says ‘Hey! Guess what? I’m also making fun of the people who abuse the Drake equation.’

    Amazingly, sometimes a joke can work on more than one level.

    And you don’t think the conclusion was meant to be a little absurd? After all, its the punchline of a joke. Or am I reading too much into it?

    Anyway, why don’t we ask Randall? He’s pretty accessible, I understand.

  14. Jon – I took exception to the argument that because people were disagreeing with you they were being defensive and that was further evidence you were right.

    You wrote: “I think you missed one of the major points of this cartoon – funny, because its a pretty direct indictment of the Drake equation itself. ”

    If you are now clarifying your position that you meant abuse of the Drake equation, I accept that.

    Of course I understand that humor can work on multiple levels. I just don’t think it was a “direct indictment” and it should be “funny” that I missed it.

    It seemed to me that the fact you think this cartoon is an indictment of the “Drake equation itself” likely stemmed from confusing criticism of the equation with criticism of abuse of the equation. Because you think that using a similar strategy is inherently a joke.

    But in fact his use of the “Flake equation” was quite reasonable, if based upon acknowledged assumptions. If criticizing the Drake equation itself was his intent, I don’t think he accomplished his goal. But that is a matter of interpretation.

    In any case, we are arguing over a subtle point that it seems we actually agree on. And sorry if I came off harsher than I intended.

  15. jblumenfeld says:

    I will cop to the ‘use-mention’ error of being imprecise, which allowed ‘Drake Equation’ to be conflated with ‘Abuse of the Drake Equation.’

    By the way, I do get your point about the ‘reasonableness’ of the Flake Equation. I just think its funny how something reasonable can be absurd in that way.

    Thank you for your post – and I’m sorry for getting a little harsh myself.

    I’m glad that’s all sorted out – and by the way, I love XKCD. It officially makes me an old man, though, because my teenage kids brought it to my attention.

  16. jblumenfeld says:

    ‘Use-mention’ error, by the way, is a classic logical fallacy that comes up in the political arena all the time. I only just learned the name of it during the healthcare debate. If you don’t have it on your list, its a good one.

  17. locutusbrg says:

    This is slightly left of topic and I am no Physicist. The point of the equation is to determine the probability of life in the universe, I believe. For the determined UFO acolytes, no argument will sway them. However when the average person hears the UFO believers quote numbers based on this equation it generates an unrealistic opnion about visitation. The probability of visitation is vastly less likely than what the numbers represent. Given super-luminal distances, relativity, and the extremely short period of time we have been around the drake equasion does nothing to support visitation ny a sophisticated life form. I think it reasonable to assume that there may not be any life right now in the universe given our point of view and relativity. From the quantum standpoint does it exist if it is not observed? Relativity and stellar distance dictates that life may have risen and fallen many times over the course of the universe, but that does not in any way mean that it is there is observable life now even if the drake equation is correct. To put it in context would you see a fire or smoke if it has only been around for a quintillionth of a second, 60 miles away. Even if an advanced civilization is looking for something like us they may not observe us given the short time we have been around. Even if they have developed some sort of superluminal travel, would we use the concord to go visit an anthill? The drake equation is tool that some well meaning, myopic people, use to justify the feeling that we are important on some level to “greater Beings”. The Drake equation has become a tool used by “flakes”, to provide the average person with an unrealistic view of alien visitation. Maintaining an illusion of likely versus very very unlikely.

  18. locutus,

    I agree that it is further potential for abuse to use the Drake equation to calculate the probability of alien visitation, because that adds more variables not in the equation. As you say, even if there are alien civilizations out there, chances are they are very far away, and interstellar travel may be forever slow and impractical.

  19. baimeeker says:

    Just for the record, Monroe has directly made fun of the Drake equation before:

    Still, had I not seen the previous comic, I would have had a much harder time seeing this comic as criticizing the Drake equation itself. It’s a really good indictment of the “it works because of my anecdote” style of thinking.

  20. superdave says:

    XKCD funny. Me like.

  21. John D. Draeger says:

    “While XKCD applies his equation to UFO sightings, it applies equally well to any paranormal or pseudoscientific phenomenon…”

    With a slight modification the equation would work for supernatural claims too. Since religious claims are more socially acceptable than claims of aliens, we can multiply the T and both F variable estimates by 10. So that’s how we arrive at our current state of having most people on the planet with at least one major delusion.

  22. DLC says:

    Consider also that people have even been known to remember things a certain way just because a skilled but careless (or malfeasant) interviewer has asked them the right leading questions in the right way. One good example of this is from the days after TWA 800 blew up. conspiracy theorist “investigators” went up and down the east coast looking for people who “Saw Something”, and, because they asked just the right question they got just the right reply. The problem is, all the (now tainted) witnesses saw the same thing at the same time in the same way.

  23. John2 says:

    Thanks for this, it’s my favourite approach to debunking the notion that the “three foot stack” of poor evidence is, when taken en mass, proof of alien visitation (i.e. by pointing out that what we see matches precisely what we’d expect if the “no visitation” hypothesis were real.)

    I also got the nice treat of the above example of one of my favourite poor web arguments that I needed to complete this week’s “bullshit Bingo”. Thankfully JBlumenfeld gave me the classic “X. I never Said X. Oh well, yes, I did say X, but I thought I’d claim otherwise despite it being still written above”.

    I wonder if anyone has done a serious study on how internet “debate” is instead reduced by so many to a points scoring game in semantics. The same thing does happen in face-to-face interactions, but in my experience only to a far lesser extent. It often seems that the web is populated by people who have never had a real intellectual debate, and who instead rely on one or two very simple pieces of rhetoric in place of genuine discussion.

    All very rum.

  24. davidsmith says:

    While I agree that biases of cognition and perception certainly occur, I am extremely sceptical of what boils down to a subjective assessment about whether the number of UFO reports, or other paranormal experience, is what we would expect given the existence of such biases. There is just too much uncertainty. The same applies for the argument about coincidence. I think it’s more a matter of faith that these kinds of explanation, which work in principle, are actually responsible for all cases. But then, Steve also added the proviso that no evidence to contrary exists, which isn’t entirely true.

  25. BillyJoe7 says:

    You guys have made this unnecessarily complicated. Damn you! I’m going to read the article again and forget about you lot. Use-mention my ass!

  26. BillyJoe7 says:


    “I am no Physicist.
    From the quantum standpoint does it exist if it is not observed?”

    I’m not a physicists either, but the answer to that question, if it is a serious one, is “no”.

  27. Dr M says:

    “I am no Physicist.
    From the quantum standpoint does it exist if it is not observed?”

    I’m not a physicists either, but the answer to that question, if it is a serious one, is “no”.

    I am a physicist, and the answer to that question, if we are to take it seriously, is that it is meaningless. Applying quantum-mechanical concepts to macroscopic objects — let alone entire species — is tenuous under the best of circumstances, and in most cases outright ridiculous, given the fact that truly macroscopic objects a) exist on scales many orders of magnitude larger than Planck’s constant (meaning that quantum effects are going to be negligibly small), and b) will suffer immediate decoherence, destroying any quantum-mechanical superposition.

    Macroscopic objects are well described by classical physics, which means exactly that it makes no sense to ask about quantum-mechanical effects. This, by the way, is what Stephen Hawking referred to when he famously said that “when someone mentions Schrödinger’s cat, I go for my gun.”

  28. BillyJoe7 says:

    Dr M,

    I was being as brief and as dismissive as possible. 😀

    I think there are three reasons and, if I may be allowed, I will rephrase the two you mentioned in my own words:

    1) Although quantum effects do occur at the macroscopic scale of ordinary everyday objects, the effect is so vanishingly small that, after jumping a wall every second for a trillions upon trillions of years, your chance of landing on the other side is still so close to zero as to be indistinguishable from it.
    2) Quantum indeterminacy applies only for as long as a quantum particle is not interacting. For a macroscopic object consisting of trillions upon trillions of quantum particles, this is a vanishingly, infinitesimally small period of time.

    But the third reason is the real killer:

    3) The “oberver” has no special place in quantum mechanics. In particular, *consciousness* plays absolutely no role at all in decoherance. In the original double slit experiments, the “oberver” was simply a “detection device” placed at the slits. A detection device is, of course, not conscious. The experimenter himself could have dropped dead after pushing the time delay button to start the experiment for all it would matter to the outcome of the experiment.

    It is simply nonsense that moon doesn’t exist if no conscious oberver is watching. It is not true, it never was true, and never will be true.


  29. jblumenfeld says:

    John2 – Not fair. I think I was pretty clear, and I also think I was big enough to say where I thought the confusion was coming from. If that qualifies for your bullshit bingo, I think you’re playing the wrong game.

    Show me with quotes, and I’ll admit I was wrong.

    And okay, now I’m being defensive.

  30. Dr M says:


    In the sprit of the post, I was taking the point from your short dismissal and beating it to death. 😉

    What you write is absolutely correct. I was expanding a little on quantum mechanics as applied to macroscopic objects (your points 1 and 2). Your third point is both true and very important to the understanding of the implications of QM, and applies to all QM considerations, macroscopic or not. Therefore, that should actually be point number 1, with the other two as a follow-up to explain why it is also ridiculous even to drag QM into macroscopic matters.

  31. BillyJoe7 says:

    Dr M,

    Thanks for your response.
    A layman is never sure he has things right, so it’s good to get some feedback from an authoritative source (not that we should believe all authoritative sources without question of course!)

    “Therefore, that should actually be point number 1”

    That is why I called it “the real killer” argument 🙂

  32. Swoopy says:


    Thank you for making a very clear assessment of an issue that continues to come up in discussions between Skeptics and Atheists, as well as Skeptics and Deists and people of other faiths.

    While I don’t believe that this issue will ever stop being a catalyst for engaging and endless discussion between folks interested in the questions posed by believers and non-believers alike to Skeptics on these matters, I think if we can assert your well thought out and rational approach, we can keep this discussion from overwhelming and overshadowing the many other topics that are important to Skepticism.

    Other prominent Skeptics continue to have to spend their time defending their position on discussing Faith or choosing not to discuss Faith alongside many of the other familiar memes, as choosing not to engage in these types of debates as seen as an attempt to deny that there is a need to discuss them at all. The biggest issue here, at least in my opinion, is that so much time is devoted to this issue – upon which I think we mostly agree – that discussions of dangerous and timely issues such as anti-vax rhetoric, dangerous homeopathy, and other topics of consumer protection and Science advocacy get lost or moved relegated to “also ran” status.

    The sooner that Skeptics can come to a blanket understanding about the relationship between ideas of Faith and Belief (you are right to point out that Religion is an altogether different matter, as is the Church as an establishment) the sooner we can ease tensions that invariably arise between Skeptics and Atheists who see us as not caring about the question, or looking to avoid it.

    Again, thank you for stating this idea so clearly. I can only hope this article will be widely disseminated.

  33. mowatk says:

    hey Steven,

    here is a pretty charming cartoon featuring the fallibility of memory, its animated by cartoonist Chris Ware using audio from This American Life. Hope you enjoy it,

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