the official site of Michael Shermer

top navigation:
Scientific American

Folk Numeracy & Middle Land

published September 2008 | comments (8)
Why our brains do not intuitively grasp probabilities, Part 1
magazine cover

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 disconfirming 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 network social 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.

Next month, in Part 2, we will consider how randomness rules our lives through the metaphor of “the drunkard’s walk,” well elucidated by physicist Leonard Mlodinow of the California Institute of Technology in his new book of the same title.

topics in this column: , ,

8 Comments to “Folk Numeracy & Middle Land”

  1. Bill Treumann Says:

    I wrote a comment here but I wanted to send a copy to my grandson so I cut and pasted it. I didn’t realize it would disappear here. I dislike writing something a second time – so how can I send it to you?
    If you don’t want your email address bandied about I promise to reveal it to no one.
    Bill Treumann

  2. Read today Says:

    […] Shermer continues his Folk … series with an easy to read installment on Folk Numeracy, covering statistics and confirmation […]

  3. Cem Egrikavuk Says:

    Well, I logged in to make a comment on the second part of this article, published October 2008. It’s not here yet, but I’ll post anyhow;

    I strongly believe there is an error in the version of the puzzle presented in the opening paragraph of “A Random Walk through Middle Land, Part 2,” otherwise a great article. Shermer declares that the host knows the contents of the three doors. Therefore Monty’s actions can not be considered “Random” in such a puzzle, and compared with dices and roulette table. The hosts action to reveal the contents of a door might have the intention of deceiving the player into giving up on his already correct choice.

    Only if the rules of the game dictate that the host must reveal one of the doors after the player makes his preliminary choice, then the argument holds. As long as opening a door is optional, the game is non-random.

    (I remember the orginal version of this puzzle had the door/box being opened at random.)

    Regards,

    Cem Egrikavuk
    Istanbul

  4. Bill Zipperer Says:

    Narative bias is another problem: people latch onto a good story. How many times have we heard of a solution to a problem that seems simple and logical yet turns out to be wrong? Science and especially statistics(!) doesn’t commonly fit into a compelling narative that people can embrace easily.

  5. Chris Says:

    Cam,

    You are correct that the host’s action cannot be considered random. However, he always opens a door (both in the show and in the puzzle). The only choice is which door. In the original version of the puzzle, as published in Atlantic, it was not a random choice: the host knew which door held the car and always revealed a goat. And that is what throws people about the result. If the opening was random, your odds would indeed be 50-50. (None of this discussion applies to the actual show, by the way, which operated differently.)

  6. Ted Says:

    In the Random Walk article, the two examples, 3 doors and 10 doors, new information is introduced as the doors are opened. This changes the game. Each time a goat is revealed, the size of the game is reduced by one, and the probabilities assigned to each door must be adjusted, but all remain 1/n, where n is the number of unopened doors. When only two doors remain unopened, it is 50/50. No asymmetry has been introduced by the goat-revealing process described in the examples.

  7. Steph Says:

    If Monty opens door number 2 and reveals a goat, then that would automatically eliminate possibility b.)bad, good, bad. So with only two possibilities, shouldn’t the odds be 50/50?

  8. Jeff Says:

    Wouldn’t this actually be a short-term/long-term thing?

    By expanding this and saying you have 10 sets of three, and while the correct options are random, your choice goes throughout all, so, while set one, door one is a goat, set two door one is a car. You choose one number, and it is applied to all sets.

    Well, opening all door number twos, wouldn’t that make the individual odds, for each set, either 50-50 or 0-100, but the long-term odds, that you will have more cars than goats, are 66-34?

    Am I making any sense?