Have you ever gone to the phone to call a friend only to have your friend ring you first? What are the odds of that? Not high, to be sure, but the sum of all probabilities equals one. Given enough opportunities, outlier anomalies—even seeming miracles—will occasionally happen.
Let us define a miracle as an event with million-to-one odds of occurring (intuitively, that seems rare enough to earn the moniker). Let us also assign a number of one bit per second to the data that flow into our senses as we go about our day and assume that we are awake for 12 hours a day. We get 43,200 bits of data a day, or 1.296 million a month. Even assuming that 99.999 percent of these bits are totally meaningless (and so we filter them out or forget them entirely), that still leaves 1.3 “miracles” a month, or 15.5 miracles a year.
Thanks to our confirmation bias, in which we look for and find confirmatory evidence for what we already believe and ignore or discount contradictory evidence, we will remember only those few astonishing coincidences and forget the vast sea of meaningless data.
We can employ a similar back-of-the-envelope calculation to explain death premonition dreams. The average person has about five dreams a night, or 1,825 dreams a year. If we remember only a tenth of our dreams, then we recall 182.5 dreams a year. There are 300 million Americans, who thus produce 54.7 billion remembered dreams a year. Sociologists tell us that each of us knows about 150 people fairly well, thus producing a social-network grid of 45 billion personal relationship connections. With an annual death rate of 2.4 million Americans, it is inevitable that some of those 54.7 billion remembered dreams will be about some of these 2.4 million deaths among the 300 million Americans and their 45 billion relationship connections. In fact, it would be a miracle if some death premonition dreams did not happen to come true!
These examples show the power of probabilistic thinking to override our intuitive sense of numbers, or what I call “folk numeracy,” in parallel with my previous columns on “folk science” (August 2006) and “folk medicine” (August 2008) and with my book on “folk economics” (The Mind of the Market). Folk numeracy is our natural tendency to misperceive and miscalculate probabilities, to think anecdotally instead of statistically, and to focus on and remember short-term trends and small-number runs. We notice a short stretch of cool days and ignore the long-term global-warming trend. We note with consternation the recent downturn in the housing and stock markets, forgetting the half-century upward-pointing trend line. Sawtooth data trend lines, in fact, are exemplary of folk numeracy: our senses are geared to focus on each tooth’s up or down angle, whereas the overall direction of the blade is nearly invisible to us.
The reason that our folk intuitions so often get it wrong is that we evolved in what evolutionary biologist Richard Dawkins calls “Middle World”—a land midway between short and long, small and large, slow and fast, young and old. Out of personal preference, I call it “Middle Land.” In the Middle Land of space, our senses evolved for perceiving objects of middling size—between, say, grains of sand and mountain ranges. We are not equipped to perceive atoms and germs, on one end of the scale, or galaxies and expanding universes, on the other end. In the Middle Land of speed, we can detect objects moving at a walking or running pace, but the glacially slow movement of continents (and glaciers) and the mind-bogglingly fast speed of light are imperceptible. Our Middle Land timescales range from the psychological “now” of three seconds in duration (according to Harvard University psychologist Stephen Pinker) to the few decades of a human lifetime, far too short to witness evolution, continental drift or long-term environmental changes. Our Middle Land folk numeracy leads us to pay attention to and remember short-term trends, meaningful coincidences and personal anecdotes.