Causality, Coincidence and occasionally expected surprises

In this post I’ll deal a bit with common misconceptions regarding the world around us and a few tips of how to worry less about your kids.


When two phenomena always coincide, what do we mortals think is happening? How do our brains process this information? Well, I believe we are almost always led to the wrong conclusions and that our intuition regarding co-incidences (as in two things happening together) and coincidences (as in two things happening together unexpectedly) is crappy.

Let me try and explain myself.

There are several distinct cases where two things (A and B) appear together.

The first commonly assumed explanation is that one of them is the cause and the other the effect, so the two options are:

A causes B

Or maybe

B causes A.

Sometimes this is true and examples are abundant.

Think of Explosion and Explosive-bang: An explosion is always accompanied by a loud bang (though a bang is usually not accompanied by an explosion). In this case we would be correct in deducing that the explosion was the cause and the bang the effect.

Now consider the case of rain and clouds: clouds sometimes produce rain, but rain is always accompanied by clouds. If we ignore the fact that clouds appear before rain, one could be led to the assumption that perhaps rain causes clouds and not the other way around. So even here, no clear cut case. Let's continue.

A common error in these cases is to swap cause and effect, which can lead to interesting results. I once read about a research, which claimed that crowded jails are a precursor to crime waves, and therefore sometimes felons should be set free in order to avoid jammed prisons. This conclusion was reached by analyzing data of jail occupancy and the number of crimes committed over a period of time. It may have looked something like this:


Though crowded prisons are indeed a mess, the truth is that after many crimes are committed, many people go to jail, and not the other way around.

So far so good, but now it gets better. Two phenomena may be correlated although they have no direct link between them. Take glacial melting in Greenland and the disappearance of certain kinds of cold-loving butterfly species from Israel. Both have been witnessed to occur recently but there does not seem to be any direct link between them. The glacial melt goes nowhere near the butterflies and it is implausible that the absence of the insects causes the ice to thaw. That’s because there is no direct link between them, they are both side-effects of global warming. So we are presented with the third kind of co-occurrence:

C causes A, and, C causes B

Whenever C is present (but may go unnoticed), A and B appear. This may be misinterpreted to give some weird and comic results like this one, which urges people to become pirates in order to stop global warming (from The Spaghetti Monster web site):

Image This inverse correlation between global warming and the number of pirates is of course due to a third phenomenon – human technological advancement. Modern technology enables better control of the oceans and at the same time releases greenhouse gasses, causing the planet to warm up.

Here is one showing a correlation (p-value < 0.01) between the number of breeding stork pairs and human births is some European countries (by Robert Matthews Aston University):


Can you guess where this correlation stems from? If you do, leave a comment with your explanation (I’ll give you a hint, storks don’t really deliver babies).

It’s worth noting a fourth option for misinterpretation – when A is actually the same as B (in a way). Consider the almost perfect correlation between a person’s age and the number of birthday parties that were given in his honor. Would it be correct to deduce that birthday parties are good for one’s health? Probably not.

I’m pretty sure that each of these mistakes happens quite frequently; never underestimate human propensity for errors.

There is yet a fifth option I would like to present, which I believe is the most frequent and the most commonly overlooked. That is the case when A and B appear coupled, but they are not at all related, not even through a third phenomenon like before.

Consider the following situation: The Chubooboo, a rare and almost extinct species of monkey is discovered in the rain forests of Congo. There are just 5 specimens left in the world. Upon closer examination, these monkeys appear fat and sluggish, hardly able to move between the trees. Researchers quickly conclude that the poor Chubooboo became extinct due to their lack of speed and agility, unable to compete with the rest of the forest dwellers.

This seems like a logical assumption. There are many cases of incompetent creatures who failed to make the Darwinian mark. The Dodo is one, being slow and flightless; it nested on the ground and could be easily caught. It and its eggs were easy prey for the sailors and the rats they brought along. As a consequence, the Dodo is no longer with us.

Another good example is the universally beloved evolutionary disaster – the Giant Panda. The Panda has failed Darwin at every possible turn, shaming evolutionary biologists and stacking insurmountable hardships on conservation efforts.

This sluggish bear is a carnivore by lineage but survives almost exclusively on bamboo. It derives so little nutrition from its food, that it must constantly eat, shuns social interaction and avoids slopes to conserve energy. The female is ready to mate 3 days once a year and her chances of finding a mate, Pandas being solitary, is slim.

So it’s not like there is no precedence.

But I’ll let you in on the truth: the average Chubooboo was a quick and lithe little ape, easily beating any of the other forest creatures to food and shelter. In fact, the Chubooboo have almost become extinct due to a new virus which has affected their ability to reproduce.

So, how come the ones left are so lame? Well, the Chubooboo, like other monkeys have between them slower and chubbier specimens from time to time. It is by chance alone, that the last 5 remaining specimen were all fat, causing scientists to jump to the wrong (albeit natural) conclusion.

Improbable you say? Well let’s assume that the probability of a Chubooboo being especially lugubrious is about 1 in 10. Then the probability of the last 5 specimens all being small lard buckets, is one in 100,000. Now, that does indeed seem quite unlikely.


Given my in-depth knowledge of Chubooboo biology, I will let you in on the following facts:

1 in 30 is gay and will not reproduce

1 in 70 is born deaf (but survives)

1 in 100 is born blind (but survives, let’s assume for argument’s sake)

1 in 200 is born a dwarf and is especially weak and slow

1 in 500 is born with short arms, rendering it practically incapable of scratching itself

1 in 1000 is born with an enlarged head, causing frequent headaches and often getting stuck between branches

Etc. etc. etc.

So, now, is it still improbable that the last 5 specimens on earth are all fat. Yes, it is! But is it likely that given the 5 specimens, and unable to learn anything about the nature of the others, that we are led to the wrong conclusion regarding their untimely extinction? The answer is yes, very much so!

In fact, considering that there are hundreds if not thousands of species becoming extinct, we are very likely to run into groups of animals, who all share the same distinct abnormality purely by chance.

This simian side-story was only an illustration of this random coincidence effect. The following array of squares, each one having a different feature (color in this case) can be thought of as a visual interpretation of this:


Nothing seems too unusual about this array, but if you happen to have only 4 of them, you may end up with these:


Which shows a clear “pattern” – they are all red.

Finding order where none exists is connected to a very interesting subject in mathematics called Ramsey Theory. Ramsey Theory broadly states, that any configuration of elements, no matter how diverse or random, if large enough, will always hold an “ordered” substructure. See here for more details. 

Mathematical “order” is a rather broad term, what it means in human terms is quite another thing – let’s give it a look.

We have a knack for perceiving order, patterns, forms and matches, even when these are not actually present. For example, here is a true case study from my marriage:

After having dated A. (who is now my wife) for several months, we stumbled upon the discovery that both our fathers were born on the same date – June 30th. My wife thought this quite a coincidence and was truly vexed when I showed no such disposition. Matters deteriorated when I claimed that these things are bound to happen and so this is no great surprise. This has since turned into a long term disagreement, which my wife utilizes to prove that I lack basic human skills such as surprise, glee, feelings and the like, and I use to claim that my wife’s intuition regarding certain combinatorial structures is flawed (I rarely run into situations where this claim can be made, but I keep it handy, just in case). 

So who’s right? Well, like any marriage counselor would tell you, we both are. Again, the explanation stems from coincidence and order in large structures:

The probability for both our fathers to have the same birthday (note, not the same day, just the same date) is rather small, 1 in 365 to be precise. You would on average, have to knock on 365 doors before you found a family with such a trait, which does make us pretty special (assuming couples have just two fathers, or that they know who the fathers are, or that there is a father, not trivial in today’s lax family structure).

In this regard, my wife is correct, as she usually is.

Except when she’s not.

Was I right to state that there is nothing to be surprised about? Let’s make a (partial) list of other coincidences which would have surprised us just as much:

ImageBut why stop at birthdays? Many family features with a certain order would amaze us: Having a sister and brother on each side with the same name, having two cousins on each side with the same name, etc. etc..

Now the thing is, that no matter how you try and arrange the names, dates and other family minutiae, you are likely to have something “in order” that is, in some human-meaningful pattern. This is exactly what Ramsey Theory says.

To strengthen this point, here is another (true) story:

When we got married, my cousin from the States came over to celebrate with us. My cousin’s wife is called Kim and she has a sister called Lisa. My wife, an avid dog lover has had two important dogs during her life; one was called Kim and the other Lisa. What a coincidence! (no, really, it is…).

To sum things up, my wife was right to be surprised by the b-day affair and I was right not to be surprised at her surprise. The probability of each coincidence is sometimes minute, thus justifying amazement, but there should be no astonishment that you are astonished once in a while. It must be that way! In fact, if you were not to come across a coincidence fairly often, you should be very surprised (and maybe even a bit worried).

Now, I promised something to abate the concern some of you parents feel regarding your children… so here goes…

About a year ago my first child was born. I found the sea of child statistics an environment rife with misunderstandings and logical aberrations. Think about this – what gets parents worried? Well, I would say the most common cause is that their kid is not on par with what other kids of his/her age do, or what the books say should be the norm at this stage.

Yeah, well, so what? (you may ask) So – one would expect most parents to be calm, since most babies are normal and only the minority to be stressed out. Right?

But this is not the case. In fact, almost all parents look at their child and see something out of the ordinary (in a bad way) and worry. How can this be?

Well, consider how many different features parents look at:

Image And there are plenty more (these are just baby traits, I don’t want to know what happens when they hit puberty).

Let’s assume there are about 30 different features parents look at during the first years. Now, let’s assume the children are normally distributed with regards to all traits (this does not have to be the case in order to prove my point, but it’s a reasonable assumption). What are the chances that you will worry about your child? Well, you should (maybe) worry if he/she is off on some parameter. How far off? Well, let’s check one standard deviation from the norm. In a normal distribution, 66% are within the first standard deviation from the norm and 94% are within two standard deviations. So the probability that you will not need to worry is the probability that your little loved one is within the norm in all terms, i.e. 0.66^30=0.00005 and the chances that you will worry about at least one feature is 1-0.000005=99.9995%, which means, almost certainly!

There is an implicit assumption here that the traits are independent, which is of course not true. A lighter baby is likely to be shorter and slimmer and maybe weaker as well. But I dare say the overall trend remains.

Even if we were to take two standard deviations as the ‘worry’ threshold, 94% of babies are within the norm in each trait, and the probability you will worry is 80% which still means most people worry and for naught. In fact, it is very improbable that any one baby will be normal in all respects. Which is actually normal. Get it?

So my friends, do not take coincidences too seriously and always consider what other configurations would have produced the same effect on you. And your baby is fine. Probably.



6 תגובות בנושא “Causality, Coincidence and occasionally expected surprises

  1. Wonderful. The first part should be required reading for all first year university students, especially for economists. I like your point at the end about child development. It's a good point to make because I think it's harder for people to realize this when focusing only on their own kids, or when dealing with chart bearing professionals.

  2. Excellent post! Nevertheless, I would add three comments:
    1. The Chubooboo should be monkey, and not ape (you use these two terms), since there are only 6 known ape species, and many monkeys.
    2. It is not logical that the last five Chubooboo specimens were all lame just by chance, since being lame is a serious Darwinian disadvantage. However, they all were very old! Before this population had lost its propagation ability they were included in the last born Chubooboo generation. Naturally, most specimens of this generation died during their life, due to whatever reasons. Only few lasted for a ripe old age. These are the five last Chubooboo monkeys. More of that, it is reasonable that these old specimens were precisely stronger than the average, both genetically (better immune system, stronger heart-lung system etc) and behaviorally (kept physical activity, didn't smoke etc). Nevertheless, their lameness may be an outcome of their old age…
    3. You've made a considerable effort to prove that parents should not worry for their child when he/she is off the SD of a specific feature. However, one may prove the opposite argument using the very same logic. It is clear that the abnormal children regarding this feature are within the 34% children that are off the SD, and for sure there are within the 6% that are off two SDs. If my child is also included there, I can't know if he is among the happy normal distributed children, or (God forbid) one of the abnormal ones. I have to worry, and use consultation. Actually the situation is even much worse, since my child my be abnormal in some feature (hurt work for instance) and still looks and behaves within the SD regarding all parent observations. Therefore a parent should be worried constantly, even (and mainly) if everything looks OK, and rash to the doctor every Monday and Thursday, to check if his child is still alive.
    Well, my mam used to act (almost) this way.

    1. Dad, thanks for the detailed comment!
      I agree with almost everything you said, the Chubooboo part could have been explained better, no doubt.
      I still do not fully agree about worrying with children. It is true that any deviation can mean trouble, but what I claim is that on average, the overwhelming amount of time, a deviation does not mean anything. So, if you were never to go to the doctor with a deviation from the norm, you would have a positive expectation of your and your kids' time and money spent.
      The only problem is, that into the calculation you need to add the negative cost of not going to the doctor when something is wrong. Sometimes (but very very rarely) this negative cost is great. The question is how much negative cost you assign to each possible outcome (kid is sick but could have been healthy, kid is permanently injured when this could have been saved, or worse…).
      You then multiply the cost by the probability and try to maximize the expectation.
      Since the cost of anything bad happening to your child, when assigned by Polish mothers, is minus infinity – the expectation is only positive if you visit the doctor two times a week. This is understandable, but statistically flawed 🙂

      Just to be honest, we have visited the doctor with Eylon about 10 times so far, for checkups and for illness. Not once, and I mean not even once, was there something wrong which did not go away by itself. We were not even once given medication or told to do something differently. So our expectation so far is very negative. We always say that the doctor is more a parent psychologist than a pediatrician. She probably does not even practice medicine at all. No need.
      I intend to do better with the next child…

      1. Oren,
        We both know that the question is neither why a parent should be never worried about his child, nor why he should always be. The real question is when he should, and when he shouldn't. I would suggest four roles for that:
        1. Parents should get knowing their specific child enough for distinguish (usually) between a normal disorder and an abnormal one.
        2. Parents should be informed and aware regarding symptoms of some common serious diseases, like pneumonia, meningitis etc.
        3. When you are not sure, prefer a positive mistake (I'm afraid my child deserves a special treatment, and the doctor found it unnecessary) over a negative one.
        4. And the most problematic one: if one of the parents asks for a doctor examination, usually do it. The parent peacefulness is important for all of you. Also, by acting this way both parents would advance towards role No. 1 (see above).
        Therefore, without knowing details, I assume you've usually behave close enough to these roles, and you personally shouldn't feel bad.
        In regarding the next child, you'll surly act differently, since every child is different, and also because you've became more experienced.

  3. To my dear cousin and his father, I have to say how much I enjoyed this blog, and would like to add some of my experience…Since my wife and I have raised 3 daughters, 2 of whom are now in college, I think we can help here…when our first one was born, we took her to the hospital within 3 says in a panic because of something or another, and were promptly told to go home, not to worry, everything's "normal"…as time went on we relaxed a bit, maybe too much as I personally dislocated 3 elbows of my first 2 children, and while my wife considered leaving me because of my roughness not to mention my immaturity, I was able to relocate the elbow on my 3rd time on my own…all said, we've had broken arms, toes, lots of stitches, bloody noses and heads, minor surgeries, and lots of drama. I can assure you that while puberty is not a fun time (especially with girls), it doesn't necessarily get much easier….you will NEVER stop worrying about them, NEVER.

    It just comes with the territory, and this parenting thing is anything but easy, but it's also the most awesome thing you'll ever do in your life as you probably already know…


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