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The Flake Equation

Estimating the number of people who have
experienced the paranormal or supernatural

The Drake Equation is the famous formula developed by the astronomer Frank Drake for estimating the number of extraterrestrial civilizations:

N = R × fp × ne × fl × fi × fc × L where…

  • N = the number of communicative civilizations,
  • R = the rate of formation of suitable stars,
  • fp = the fraction of those stars with planets,
  • ne = the number of earth-like planets per solar system,
  • fl = the fraction of planets with life,
  • fi = the fraction of planets with intelligent life,
  • fc = the fraction of planets with communicating technology, and
  • L = the lifetime of communicating civilizations.

The equation is so ubiquitous that it has even been employed in the popular television series The Big Bang Theory for computing the number of available sex partners within a 40-mile radius of Los Angeles (5,812). My favorite parody of it is by the cartoonist Randall Munroe as one in a series of his clever science send-ups, entitled “The Flake Equation” (on for calculating the number of people who will mistakenly think they had an ET encounter.

Such multiplicative equations for calculating the product of an increasingly restrictive series of fractional values are effective tools for making back-of-the-envelope calculations to solve problems for which we do not have precise data. To that end I thought it a useful addition to the Skeptic toolbox to create a Flake Equation for all paranormal and supernatural experiences (and in the Flake Equation I’m interested not in beliefs but in actual experiences that people report and that we hear about, because this becomes the foundation of paranormal and supernatural beliefs):

N = Pw × fp × fm × ft × nt × no × fm where…

  • N = Number of people we hear about who report having experienced a paranormal or supernatural phenomena,
  • Pw = Population of the United States (January 1, 2012: 312,938,813),
  • fp = Fraction of people who report having had an anomalous psychological experience or witnessed an unusual physical phenomena (1/5),
  • fm = Fraction of people who interpret such experiences and phenomena as paranormal or supernatural (1/5),
  • ft = Fraction of people who tell someone about their experience (1/10),
  • nt = Number of people they tell (15),
  • no = Number of other people told the story by original hearers (15), and
  • fm = Fraction of such stories reported in the media or on Internet blogs, tweets, and forums (1/10).

N = 28,164,493, or about 9 percent of the U.S. population.

To compute this figure I used the 2005/2007 Baylor Religion Survey, which reports that

  • 23.2% say that they have “witnessed a miraculous, physical healing,”
  • 16.3% “received a miraculous, physical healing,”
  • 27.5% “witnessed people speaking in tongues at a place of worship,”
  • 7.7% “spoke or prayed in tongues,”
  • 54.5% experienced being “protected from harm by a guardian angel,”
  • 5.9% “personally had a vision of a religious figure while awake,”
  • 19.1% “heard the voice of God speaking to me,”
  • 26.1% “had a dream of religious significance,”
  • 52% “had an experience where you felt that you were filled with the spirit,”
  • 22.1% “felt at one with the universe,”
  • 25.7% “had a religious conversion experience,”
  • 13.8% “had an experience where you felt that you were in a state of religious ecstasy,”
  • 14.2% “had an experience where you felt that you left your body for a period of time,”
  • 40.4% “had a dream that later came true,” and
  • 16.7% “witnessed an object in the sky that you could not identify (UFO).”

This works out to an average of 24.4 percent, thereby justifying my conservative 20 percent figure for fp and fm. The other numbers I gleaned from research on gossip and social networks, conservatively estimating that 10 percent of people will tell someone about their unusual experience, and that within their average social network of 150 people they will tell at least 10 percent of them (15) who in turn will pass on the story to 10 percent of their social network of 150 (15). Finally, I estimate that 10 percent of such stories will be reported in the media or recounted in blogs, tweets, forums, and the like.

Of course the final figure for N will vary considerably depending on what numbers are plugged into the equation, but the result will almost always be a number in the tens of millions, which goes a long way toward explaining why belief in the paranormal and supernatural is so ubiquitous. Experiencing is believing!

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Gambling on ET

How to compute the odds that claims of extraterrestrial life discovery are real and reliable

The Search for Extra-Terrestrial Intelligence (SETI) has to be the most interesting field of science that lacks a subject to study. Yet. Keep searching. In the meantime, is there some metric we can apply to calculating the probability and impact of claims of such a discovery? There is.

In January, 2011 the Philosophical Transactions of the Royal Society published 17 articles addressing the matter of “The Detection of Extra-Terrestial Life and the Consequences for Science and Society,” including one by Iván Almár from the Hungarian Academy of Sciences and Margaret S. Race from the SETI Institute, introducing a metric “to provide a scalar assessment of the scientific importance, validity and potential risks associated with putative evidence of ET life discovered on Earth, on nearby bodies in the Solar System or in our Galaxy.” Such scaling is common in science—the Celsius scale for temperature, the Beaufort scale for wind speed, the Saffir-Simpson scale for hurricane strength, and the Richter scale for earthquake magnitude. But these scales, Almár and Race argue, fail to take into account “the relative position of the observer or recipient of information.” The effects of a 7.1 earthquake, for example, depends on the proximity of its epicenter to human habitations.

An improvement may be found in the Torino Scale that computes the likelihood of an asteroid impact and the risk of its potential damage—from 1, a near miss with no danger, to 10, certain impact with catastrophic consequences. But Almár and Race note that “the scale does not include any consideration of the observations’ reliability.” Building on SETI’s Rio Scale for evaluating the effect on society of an ET discovery, Almár and Race propose the London Scale that multiplies Q x δ, where Q (scientific importance) is the sum of four parameters:

  • life form (1–5, from Earth-similar life to completely alien),
  • nature of evidence (1-6, from indirect biomarkers
    to obviously organized complex life),
  • type of method of discovery (1–5, from remote sensing
    to return mission sample), and
  • distance (1–4, from beyond the Solar System to on Earth).

This sum is then multiplied by δ (a reliability factor) ranging from 0.1–0.5, from probably not real to highly reliable. The maximum Q can be is 20 x .5 = 10.

For example, Almár and Race compute the odds that the Allan Hills 84001 Martian meteorite contains alien life as (2+2+4+4)0.3 = 3.6 for scientific importance and credibility, noting that “several scientific counter-arguments have been published and the discovery has not been generally accepted.” I would assess the recent claim of arsenic-based life in Mono Lake as (2+1+4+4)0.2 = 2.2, fairly low by comparison.

Such scientific scales attempt to bring some rigor and reliability to estimates of events that are highly improbable or uncertain. The process also reveals why most scientists do not take seriously UFO claims. Although the first two categories would yield a 5 and a 6 (completely alien and complex life) and its distance is zero (4, on Earth), the method of discovery is highly subjective (perceptual, psychological) and open to alternative explanation (1, other aerial phenomena) and the reliability factor δ is either obviously fake or fraudulent (0) or probably not real (0.1), and so Q = (5+6+1+4)0.1 = 1.6 (or 0 if δ = 0).

The Phoenix lights UFO claim, for example, was a real aerial phenomena witnessed by thousands on the evening of March 13, 1997. UFOlogists (and even Arizona governor Fife Symington) claim it was extraterrestrial, but what is δ for this event? It turns out that there were two independent aerial events that night, the first a group of planes flying in a “V” formation at 8:30 that started a UFO hysteria and brought people outdoors with video cameras, which then recorded a string of lights at 10:00 that slowly sank until they disappeared behind a nearby mountain range. These turned out to be flares dropped by the Air National Guard on a training mission. Ever since, people have conflated the two events and thereby transmogrified two IFOs into one UFO. So δ = 0 and Q shifts from 1.6 to 0, which is how much confidence I have in UFOlogists until they produce actual physical evidence, the sine qua non of science.

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Will E.T. Look Like Us?

Evolution helps us imagine what aliens might be like
magazine cover

What are the odds that intelligent, technically advanced aliens would look anything like the ones in films, with an emaciated torso and limbs, spindly fingers and a bulbous, bald head with large, almond-shaped eyes? What are the odds that they would even be humanoid? In this YouTube video, produced by Josh Timonen of the Richard Dawkins Foundation for Reason and Science, I argue that the chances are close to zero. Richard Dawkins himself made this interesting observation in a private communication after viewing it:

I would agree with [Shermer] in betting against aliens being bipedal primates, and I think the point is worth making, but I think he greatly overestimates the odds against. [University of Cambridge paleontologist] Simon Conway Morris, whose authority is not to be dismissed, thinks it positively likely that aliens would be, in effect, bipedal primates. [Harvard University biologist] Ed Wilson gave at least some time to the speculation that, if it had not been for the end-Cretaceous catastrophe, dinosaurs might have produced something like the attached [referring to paleontologist Dale A. Russell’s illustrated evolutionary projection of how a bipedal dinosaur might have evolved into a reptilian humanoid].

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