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# Why Our Brains Do Not Intuitively Grasp Probabilities

Part one of a series of articles on the neuroscience of chance

Image: Matt Collins

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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.

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1. 1. mattcron 11:20 PM 8/22/08

I think this article accurately portrays the fascinating crossroads of social v. hard sciences. The ability for us to equate, in our mind, improbable with impossible (or inevitable, for that matter) speaks to this notion of confirmation bias, self fulfillment. The introduction of time into the equation is new to me, and very salient. Time, place, and self or ego contribute. The quantification of relationships and networks does well here. It is painfully easy to discard the metrics of everyone else, treating yourself as the sole recipient of the worlds' stimuli -- existing in a vacuum. It is short sightedness staring into a mirror. It is particularly amazing how, in the midst of hypersensitive consciousness of self, of hear and now, that we are so obtusely disregarding of exactly that - hear and now...

http://autodidacticdropout.wordpress.com/

2. 2. skinnybear 10:04 AM 8/28/08

Right on. So what can we do about our "middle" sightedness? Perhaps a few subtle daily reminders of the scope of things could help keep our awareness at a universal level, say, a billboard with a spiral galaxy on it, or a surfer charging through subatomic particles. Undoubtedly, it is important to remind ourselves of the interconnectedness of all things large and small.

3. 3. benrast 08:42 AM 9/3/08

On the other hand, middle sight appears to have been very useful for our species. As Chesterton said, no rational man would ever marry or volunteer to fight a war.

Perhaps the task is not so much freeing ourselves from folk wisdom as it is knowing when and where to use it.

4. 4. cnemelka 11:19 AM 9/3/08

This is an excellent example of "bad science". Dubious data (5 dreams per night on average? Where are your sources for this data?)

Substantial logic and other flaws (complex random "events" are simplified down to precisely occurring "bits").

I'm sorry. I'm not going to spend any more time on this.

5. 5. cnemelka in reply to mattcron 11:22 AM 9/3/08

Wow, you have an amazing intellect.

6. 6. jowilkin in reply to cnemelka 10:04 PM 9/3/08

I agree cnemelka. First of all, 1 bit a second of incoming information is rediculously low. Second, the suggestion that there would be a connection between the number of bits we take in and the probabilistic estimates we make is pretty rediculous.

Also, in the first paragraph where the author mentions that the sum of all probabilities must be one, he provides no explanation of how this should be related to the inability to estimate small probabilities as he goes on to suggest.

7. 7. jowilkin in reply to cnemelka 10:06 PM 9/3/08

8. 8. Jonarts 01:09 PM 9/4/08

Our brains are pretty good at grasping probabilities, but only when the probabilities are in the 10% to 90% range. In this range our estimated probabilities are close to Bayesian, that is, the best possible. In the tail ends, Bayesian probabilities are difficult to establish and our brains become unreliable for probability estimation.

9. 9. jowilkin in reply to Jonarts 07:20 PM 9/4/08

That is an abuse of the word "Bayesian", it generally refers either to the use of Bayes rule for inference and updating or to a view of probability as a degree of belief. But your misuse goes nice nicely with the pseudoscience in the article.

10. 10. spiralsun1 07:11 AM 9/5/08

I used to discount low-probability events until they began happening almost constantly all day long... If you really pay attention (no small feat) you will notice lots of different lines of information converging on astronomically unlikely events. I have an excellent intuitive grasp of probability and the events you cite in the article would not even be noticed because of this. The type of event I am talking about, I worked into an actual learnable methodology of analysis in my book "The Textbook of the Universe: the Genetic Ascent to God". In it, I use intropective analysis of meaning and information to demonstrate that the world has meaning built into it. The universe itself has "linguistic" qualities, and this is why we were able to develop language in the first place. There is a deeper level of meaning whereby when multiple data streams converge (many examples in book) you can no longer just calculate one set of probabilities and dismiss the event. This is somewhat like tossing a handful of coins and having them all land heads up. The probabilities are much lower for that. If I was talking to someone about heads, for example, and someone nearby hit their head on a low branch and dropped a sack of coins which then landed all heads up.... that sort of thing. That is not a real example, but it is the type of multiple event strings I use in my book to illustrate a property of the universe in general which can simplify our understanding of it and our place in it, and solve some serious mind-brain problems like the "binding problem".

11. 11. plehpamer 11:38 AM 9/6/08

A miracle is not an event with million to one odds of occurring. It is an observable event or effect in the physical or moral order where a law of nature has been altered or suspended with reference to God. A miracle is unbelievable and remains inexplicable after every possible natural explanation has been tried. If it were believable after analysis, it would not be a miracle. I think miracles are the "black swans" that falsify philosophical materialism.

12. 12. mandehu 12:25 PM 9/7/08

This article makes no sense for me: probabilities, as defined nowadays (since Kolmogoroff), are merely *some* measure of our ignorance -- not estimates of "happenability" or some such related notion. Of course I have no definition of the above :- ( To find one would be a major step forward in epistemology...

13. 13. gzuckier 08:26 PM 9/7/08

umm... wouldn't a miracle be something with precisely zero probability of occurring?

14. 14. plehpamer in reply to gzuckier 08:58 AM 9/9/08

If an event had a probability of precisely zero and then happened, we would conclude either (1) the observer is mistaken, either confused, lying, perhaps deceived or (2) the event really doesn't have a zero probability, even though we believe it does, and we have evidence that our view of reality is limited.

15. 15. Shalom Freedman 01:47 PM 9/9/08

I do not exactly get what this piece is trying to disprove. Is it that sometimes we have intuitions which are confirmed by experience, and this is to us some kind of evidence of direction, purpose, guiding- hand in our own personal lives?
Is the whole argument here in other words intended to be against some kind of religious concept of 'design and pattern' in our life?

16. 16. Shalom Freedman 01:51 PM 9/9/08

I did not get exactly what this article wanted to prove. It seemed to me it was aiming at proving that our intuitions of purpose and design in connections between events in our own personal experience is mistaken i.e. It seems to me it wishes to undermine any idea of religious meaning and purpose , Divine direction in our lives.

17. 17. jmanheim 08:08 PM 9/9/08

One of the most important factors in the inability of the brain to grasp probabilities is that, necessarily, each of us is ego-centered. I don't feel it when you stub your toe and, unless I am the winner, I do not get my fair share of the lottery. Accordingly, everyone has a circle of importance that begins with the individual and decreases as the circle spreads to groups with diminishing impact on that individual. In all cases the enemies are the laws of probability and physics (which includes all science). The defense against these adversaries is to invent games in which we, individually, expect special treatment. These include all fair gambling games with no house cut (probability of winning = probability of losing), all unfair gambling games such as slots, Vegas card games, drawings for cars, money, etc.. Prayer, of course, is the biggest lottery of all, the lottery n the sky, and people keep doing it because of the "miraculous" ground rules. If Sue survives the operation we thank God, if not we blame Nature (whoever that is) or Satan or the doctors; whatever, God never gets blamed.. There are other important factors, not the least of which is that reality is often counter-intuitive (see the Monty Hall problem for an easy example).

18. 18. jmanheim 08:09 PM 9/9/08

One of the most important factors in the inability of the brain to grasp probabilities is that, necessarily, each of us is ego-centered. I don't feel it when you stub your toe and, unless I am the winner, I do not get my fair share of the lottery. Accordingly, everyone has a circle of importance that begins with the individual and decreases as the circle spreads to groups with diminishing impact on that individual. In all cases the enemies are the laws of probability and physics (which includes all science). The defense against these adversaries is to invent games in which we, individually, expect special treatment. These include all fair gambling games with no house cut (probability of winning = probability of losing), all unfair gambling games such as slots, Vegas card games, drawings for cars, money, etc.. Prayer, of course, is the biggest lottery of all, the lottery n the sky, and people keep doing it because of the "miraculous" ground rules. If Sue survives the operation we thank God, if not we blame Nature (whoever that is) or Satan or the doctors; whatever, God never gets blamed.. There are other important factors, not the least of which is that reality is often counter-intuitive (see the Monty Hall problem for an easy example).

19. 19. wznja in reply to gzuckier 08:36 PM 9/9/08

Bravo

20. 20. wznja 08:37 PM 9/9/08

bravo

21. 21. atulgogtay 12:37 AM 9/10/08

I think even the global warming that is taking place is one of similar periodically occuring phenomena that happen over a geological time scale. I don't think it is due to CO2 emissions or the like.

22. 22. wesleyneo 01:50 AM 9/10/08

"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."

This describes the predicament public companies get into due to the pressures of meeting share holder expectations. Drastic measures, beneficial in the long term but which might cause loses or failure to meet revenue estimates for a few quarters, might not be taken up for the same reason.

23. 23. wesleyneo 01:51 AM 9/10/08

"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."

This describes the predicament public companies get into due to the pressures of meeting share holder expectations. Drastic measures, beneficial in the long term but which might cause loses or failure to meet revenue estimates for a few quarters, might not be taken up cause the management fears the market.

24. 24. wesleyneo in reply to wesleyneo 04:16 AM 9/10/08

This comment needs to be deleted. Why am I not able to find a Delete button?

25. 25. differential 07:21 AM 9/12/08

Nonsense. The reason most people think poorly about problems in probability is that that probability requires MATH----concepts that
are are actually subtle, and calculation, and tools such as algebra
and calculus. Things like the formulae of conditional probability.

The author implicitly assumes that somehow, if we had evolved
in a different environment, we would have all that somehow wired
in, without having to use our intelligence to reason it out over centuries. At best, this is wildly speculative fantasy.

In the same way, most people have strange ideas about even the mechanics and dynamics --aka physics--of ordinary objects. Ask
most people if a bowling ball can be sent in a curved path by a curved initial motion; or if a bomb falls, from a moving plane, onto
a target just below the plane at the moment of release. You will get wrong answers often--yet, we evolved dealing with motion
, throwing and dropping objects when moving.

Why? Because physics is subtle and requires math--aka calculus, algebra--and CONCEPTS--to understand. It required
centuries of deep thought and careful observation to formulate
Newton's laws--only really understood as a differential equation.
It required the invention of calculus.

People try to organize experience into mental schema--without the
careful ( aka: theorems and proofs) methods of math, and to me,
the real miracle is that they get ANYTHING right.

26. 26. Mithremakor in reply to jowilkin 02:24 PM 9/12/08

I think you may be confusing Shermer's 'bits' of information with binary bits. If Shermer is referring to the smallest quantity of information which may be usefully assimilated his number is probably close. Since the whole thing is a thought experiment meant to put our 'intuitive' grasp of probability into perspective his numbers needn't be precise at all. If they fall within an order of magnitude of whatever the true number may be they are close enough to make his point. Keep in mind that Shermer is presenting a logical argument, not the results of a scientific study.

27. 27. Mithremakor in reply to atulgogtay 02:34 PM 9/12/08

I think if you look at the data, especially CO2 concentrations in recent time as compared to those of past warming episodes, you'll find there is scant reason to believe this warming period is anything like those which have occurred in the past.

28. 28. jraba 05:31 PM 9/12/08

The theory put out by Dawkins and Sherner is just as weak and as much a guess as any other theory out there. This is oversimplification and stereotyping at its worst. I know of no one that I am acquainted with that suffers from "folk numeracy". I especially liked the term "Middle World". The explanation supporting that term actually made me laugh. These people have built up a mental construct that has the effect of limiting true unbiased scientific inquiry. What is a shame is that many people will follow them right off the cliff without verifying for themselves whether a single iota of what they are saying is merely biased opinion, or scientific fact. Looks to me like some one is trying to sell some of their own books. Unexplained phenomenon exist and the scientific literature documents this well. There is much more to this story than these guys would have you think. Instead of talking down to people who have had paranormal experiences we should instead be looking into this phenomena with a true unbiased eye.

29. 29. jraba 05:32 PM 9/12/08

The theory put out by Dawkins and Sherner is just as weak and as much a guess as any other theory out there. This is oversimplification and stereotyping at its worst. I know of no one that I am acquainted with that suffers from "folk numeracy". I especially liked the term "Middle World". The explanation supporting that term actually made me laugh. These people have built up a mental construct that has the effect of limiting true unbiased scientific inquiry. What is a shame is that many people will follow them right off the cliff without verifying for themselves whether a single iota of what they are saying is merely biased opinion, or scientific fact. Looks to me like some one is trying to sell some of their own books. Unexplained phenomenon exist and the scientific literature documents this well. There is much more to this story than these guys would have you think. Instead of talking down to people who have had paranormal experiences we should instead be looking into this phenomena with a true unbiased eye.

30. 30. jraba in reply to atulgogtay 02:23 AM 9/13/08

You have got to be kidding me. Yes there are cyclical global temperatures. The temperature increases today, however, exactly mirror carbon emissions. The increases we have now are not a natural anomaly. Stop listening to Rush.

31. 31. Science Tim 11:13 AM 9/20/08

In the Skeptic section of the October issue the "let's make a deal", Monty Hall probabilities are clearly wrong. I generated thousands of cases picking one of 10 doors using the function "=RANDBETWEEN(1,10)" to place a prize behind one door. I picked door number 1 as my choice and then discarded any case where door from 2-9 contained the prize. From the article I would then be left with a 90% chance of winning when changing my pick to door number 10. Yet this was not the case and changing my pick from door #1 to door #10 resulted in a 50% chance of winning. Try it yourself.

32. 32. cneese in reply to Science Tim 02:37 PM 9/24/08

Science Tim:
The Monty Hall probabilities are not wrong, but the explanation is far from clear. It is crucial to understand that the host always shows a goat behind door that wasn't chosen. Probably everyone understands the three possible configurations enumerated by the author: the car can be behind door one, two, or three. What is not clear from the author's explanation is that if the car had been behind door two, then the host would have shown the goat behind door three. Thus you win 1/3 of the time if you stay with door one but 2/3 if you switch. Furthermore, the second explanation given by the author is useless if you are confused by the question.

It would have been wise to site the web page covering the original controversy. (http://www.marilynvossavant.com/articles/gameshow.html)

In both the "Parade" magazine version and Michael Shermer's version the problem is presented as a leading question: You have chosen door one and the host has shown the goat behind door two. By drawing attention to a specific example we are lead away from considering all the scenarios and the rules of the game. When Marilyn vos Savant's first answered this question, here answer was correct but her explanation was:
"Here's a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You'd switch to that door pretty fast, wouldn't you?" At least some of the heat she received from academics was due to the fact that this explanation didn't help them understand the problem and solution. What I find fascinating is that she continued to receive heat after a second explanation and apparently didn't get much support until she advocated an experimentalist's approach to the problem. However, we have to factor in who was motivated enough by the controversy to write a letter. Thus, any conclusions drawn from Marilyn vos Savant's account must be acknowledged as anecdotal.

In the end, I have to ask a simple question: Isn't Shermer's premise of folk numeracy undermined by his reliance on anecdotes to make his point? Clearly people can and do commit statistical fallacies. However, the fact that we make these mistakes is not evidence for a flaw in our intuition that we have inherited from our Paleolithic ancestors.

33. 33. maltstatt 09:39 PM 9/28/08

Comment on "A Random Walk through Middle Land" Oct 2008

Dr. Shermer
Is this a test ? One chance in two is 50-50, no matter the history. Note that Monty just eliminated option (2), and left (1) and (3). Still 50-50. The same goes for the nine of ten example. The final result is still 50-50.
Marvin Altstatt

34. 34. waluigi in reply to plehpamer 06:54 PM 8/17/09

Well spoken, plehpamer. however, (3) It is possible that the probability was once zero but is no longer due to changes in environment (that we probably don't want to see happen) and we just didn't know it yet.
Also, another criteria for a miracle should be that it is good, or that it saves someone's life in some weird way.

35. 35. waluigi in reply to mattcron 06:56 PM 8/17/09

You were saying?

36. 36. waluigi in reply to benrast 07:06 PM 8/17/09

Perhaps you're right. And perhaps you're wrong. I'm not really sure of anything except that I've got a huge ugly summer science packet due Sept. 1st and I'm gonna have a heck of a time finishing it on time. It's called summer vacation, science-packet-designing idiots!

No, you shut up!

37. 37. lagrsm 09:19 AM 9/8/09

The concept of middle land, middle sight, etc is interesting and basically points out that in the observations discussed, there is an observer and that that observer has a perspective. The fact that our perspective is calibrated to best fit the real everyday uses we have for our senses and not for the interpretation of dreams or the viewing of atoms is a good thing. Ultimately our senses are tools for gathering information. What we do with that information is then relevant or meaningful. In the case of folk numerology, the use of statistical analysis helps clarify how we should accurately interpret what our senses tell us. Overall an enlightening article and a good "everyday" use of statistical analysis.

38. 38. macmillz100 08:56 AM 12/16/09

i thougth this article was informative but there could have been more real life examples and not just a bunch a data. Overall I enjoyed reading the article.

39. 39. dewittch 08:14 PM 4/4/10

I am in a STA 215 class and its amazing how some of the stuff we are learning in class was part of this article.� I really didn't STA 215 would be very interesting, but by looking at it this way it really makes you think.�

40. 40. dewittch 08:17 PM 4/4/10

I think this article was very interesting. I thought it was very cool how I could connect things I am learning in my STA 215 class to things in this article. Honestly, I thought STA 215 was going to be pretty boring, but when I look at it like this it really intrests me.

41. 41. dewittch in reply to macmillz100 08:20 PM 4/4/10

I agree with macmillz100, this article could most definiately be a lot more interesting if there were more real life examples to relate to. I think this would have made it easier to keep up with what the article is trying to say.

42. 42. cotterke 11:22 AM 9/5/10

This author does a great job of using statistics to get his point across and showing how shortsighted the population as a whole can be. The problem that comes into play is where are these statistics coming from? Are they the average of a large group or a small group? How could they possible come up with the numbers for the bits of information that we take in each day with everyone being at different stages in their lives and with all of our experience being different. If the information changes only slightly his probabilities would be very different. If Michael wants to talk about people as a whole being Middle Landers and not looking at the big picture I would like to tell him to look at the big picture. Your stats make no sense and you cannot base the population of the world off of a small sub-group that you based your information on. Randomly assigning numbers to events does not make them true or show correct probabilities that they will happen. I prefer to have faith and believe in my miracles not find the probability of them happening.

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