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

N = R × f_p × n_e × f_l × f_i × f_c × L where…

• N = the number of communicative civilizations,
• R = the rate of formation of suitable stars,
• f_p = the fraction of those stars with planets,
• n_e = the number of earth-like planets per solar system,
• f_l = the fraction of planets with life,
• f_i = the fraction of planets with intelligent life,
• f_c = 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 xkcd.com) 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 = P_w × f_p × f_m × f_t × n_t × n_o × f_m where…

• N = Number of people we hear about who report having experienced a paranormal or supernatural phenomena,
• P_w = Population of the United States (January 1, 2012: 312,938,813),
• f_p = Fraction of people who report having had an anomalous psychological experience or witnessed an unusual physical phenomena (1/5),
• f_m = Fraction of people who interpret such experiences and phenomena as paranormal or supernatural (1/5),
• f_t = Fraction of people who tell someone about their experience (1/10),
• n_t = Number of people they tell (15),
• n_o = Number of other people told the story by original hearers (15), and
• f_m = 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 f_p and f_m. 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!

In science there is arguably no more suppositional formula than that proposed in 1961 by radio astronomer Frank Drake for estimating the number of technological civilizations that reside in our galaxy: N = R fp ne fl fi fc L

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

Although we have a fairly good idea of the rate of stellar formation, a dearth of data for the other components means that calculations are often reduced to the creative speculations of quixotic astronomers. Most SETI (search for extraterrestrial intelligence) scientists are realistic about the limitations of their field; still, I was puzzled to encounter numerous caveats about L, such as this one from SETI Institute astronomer Seth Shostak: “The lack of precision in determining these parameters pales in comparison with our ignorance of L.” Similarly, Mars Society president Robert Zubrin says that “the biggest uncertainty revolves around the value of L; we have very little data to estimate this number, and the value we pick for it strongly influences the results of the calculation.” Estimates of L reflect this uncertainty, ranging from 10 years to 10 million years, with a mean of about 50,000 years.

Using a conservative Drake equation calculation, where L = 50,000 years (and R = 10, fp = 0.5, ne = 0.2, fl = 0.2, fi = 0.2, fc = 0.2), then N = 400 civilizations, or one per 4,300 light-years. Using Zubrin’s optimistic (and modified) Drake equation, where L = 50,000 years, then N = five million galactic civilizations, or one per 185 light-years. (Zubrin’s calculation assumes that 10 percent of all 400 billion stars are suitable G- and K-type stars that are not part of multiples, with almost all having planets, that 10 percent of these contain an active biosphere and that 50 percent of those are as old as Earth.) Estimates of N-range wildly between these figures, from Planetary Society scientist Thomas R. McDonough’s 4,000 to Carl Sagan’s one million.

I find this inconsistency in the estimation of L perplexing because it is the one component in the Drake equation for which we have copious empirical data from the history of civilization on Earth. To compute my own value of L, I compiled the durations of 60 civilizations (years from inception to demise or the present), including Sumeria, Mesopotamia, Babylonia, the eight dynasties of Egypt, the six civilizations of Greece, the Roman Republic and Empire, and others in the ancient world, plus various civilizations since the fall of Rome, such as the nine dynasties (and two republics) of China, four in Africa, three in India, two in Japan, six in Central and South America, and six modern states of Europe and America.

The 60 civilizations in my database endured a total of 25,234 years, so L = 420.6 years. For more modern and technological societies, L became shorter, with the 28 civilizations since the fall of Rome averaging only 304.5 years. Plugging these figures into the Drake equation goes a long way toward explaining why ET has yet to drop by or phone in. Where L = 420.6 years, N = 3.36 civilizations in our galaxy; where L = 304.5 years, N= 2.44 civilizations in our galaxy. No wonder the galactic airways have been so quiet!

I am an unalloyed enthusiast for the SETI program, but history tells us that civilizations may rise and fall in cycles too brief to allow enough to flourish at any one time to traverse (or communicate across) the vast and empty expanses between the stars. We evolved in small hunter-gatherer communities of 100 to 200 individuals; it may be that our species, and perhaps extraterrestrial species as well (assuming evolution operates in a like manner elsewhere), is simply not well equipped to survive for long periods in large populations.

Whatever the quantity of L, and whether N is less than 10 or more than 10 million, we must ensure L does not fall to zero on our planet, the only source of civilization we have known.