The Amazing Meeting 2014

Like it? Share it!

Sign up for news and updates!






Enter word seen below
Visually impaired? Click here to have an audio challenge played.  You will then need to enter the code that is spelled out.
Change image

CAPTCHA image
Please leave this field empty

Login Form



Understanding Coincidence PDF Print E-mail
Swift
Written by Kyle Hill   

Many people have a legitimate fear of numbers, equations, and probability. This “math anxiety” keeps much of the lay public from ever willfully learning about mathematics; indeed, ignorance in this regard is often touted. Commonly used phrases like “I’m not a numbers person” and “I hate math” betray that fact that a good portion of society does not understand math and consciously avoids it.

Comprehending this deficit and doing something about it should be taken up within our school system; we should engage students with math early, often, and more rigorously.

But mathematical illiteracy plays a role in perpetuating not just equation ignorance, but pseudoscience. Not understanding just how much of your life is governed by randomness generates many a fallacious belief about the way that the world works. It should be clearly understood that randomness creates coincidence. That is to say, if there were no coincidences in life, we could speculate that some outside force is controlling the events in our lives. However, with true randomness comes the expectation that coincidences will happen: there will be cancer clusters, your friend will call you just when you were thinking about them, and last night’s dream will have somehow “predicted” the events of the following day. It is with the last example, predictive dreams, which I would like to press on with. With a short lesson in randomness and probability, we can see that so-called predictive dreams (and any other event “too amazing to be a coincidence”) are nothing more than random happenings. You don’t have ESP, it’s not fate, and it’s not magic.

“I Dreamt This Would Happen!”

The purpose of this example is to show that many pseudoscientific ideas about the way the universe works are driven by a misunderstanding of randomness and probability. While predictive dreams are harmless, I would suspect that this belief characterizes the kind of thinking that underlies pseudosciences like astrology, ESP, and parapsychology.

Let’s overcome our math anxiety with a dreaded word problem. Let’s stipulate that the chance of a dream to some extent matching the events of the following day is 1 in 10,000. This means that out of 10,000 dreams, the vast majority, 9,999, will not match any future events. Let’s also assume that having a non-matching dream one night will not affect the dream of the next night, so each night is independent from one another. So given these stipulations, the odds of having a dream that does not match any real life event is 9,999/10,000. When people speak about predictive dreams, it is not as though they have them every night. If this were happening, we might consider it to be more than coincidence. However, anyone who has experienced this phenomenon (myself included) will probably tell you that they do not hit a homerun every night. It is this fact, that an amazingly serendipitous event only happens once in a while, that alludes to chance as the rational explanation.

Remembering the odds above, the chance of having a dream that does not match any real life event for two nights in a row will follow the multiplication principle of probabilities, meaning that the probability is (9,999/10,000)*(9,999/10,000). Likewise, the probability that you will have a dream that does not predict anything for three nights in a row is (9,999/10,000)*(9,999/10,000)*(9,999/10,000). Following this principle, the chance that you will have successive dreams that do not match reality can be expressed as (9,999/10,000)N, where N is the number of nights. As I said above, I don’t think that anyone would say that these predictions are a common occurrence, so let’s consider a time period of one year. The probability that you will have successive dreams every night for a year that do not predict anything would be (9,999/10,0009)365, with N equal to the number of days in a year. This results in a 96.4 percent chance that people who dream every night of a year with not have any predictive dreams. This of course means that over a period of one year, 3.6% of people who dream every night will have at least one dream that matches reality in some way. Consider that for a moment. Even though coincidences like these can drive people to believe in fate, precognition, ESP, etc., using our definition here we can say that these probabilities in large population would produce literally millions of predictive dreams each year! Even if we relax our standards and make a predictive dream a one-in-a-million event, it would still produce thousands upon thousands of predictive dreams each year by chance alone.

It’s not magic, it’s not fate, it’s not a spiritual connection with someone else; if there’s a likelihood that something will happen, however small, it is explained by chance alone that it is bound to happen to some people at some time. Look at what happened with the supposedly prophetic Nostradamus. He threw out a claim that had to do with two towers coming down and hundreds of years later something similar happened. Somehow this passes for incredible predictive powers. Knowing what you now know about randomness and large numbers, what do you think the chances are that if I through out a vague claim, something similar to that claim will happen in the next 500 years? Would that be evidence of a magical precognition on my part, or is it just randomness? The same goes for the pseudoscientists who claim to predict natural disasters. Don’t make them famous for saying that they predicted a major earthquake to happen somewhere, there’s a probability that they will guess correctly by chance alone. If they throw out enough random predictions, the statistics say that it is bound to match a few times.

What are the Odds?

Of course, saying there are odds that you will “predict” some event tomorrow with your dream tonight does not deal with the fact that you seemingly foresaw something. But this too can be dealt with. Consider the fact that what you “see” in your more bizarre dreams never comes true. You never see a great white shark with braces or are suddenly chased down the street by a cheeseburger. Only the things that are already within the realm of possibility have a possibility of happening. Let’s say that you have a dream that your friend will get in a car accident. Given how many car accidents occur, this argument falls to the same reasoning that we dealt with above: in a large population, statistically unlikely events will happen all the time (and car accidents are much more likely than you may think). Fine then, what if you dream that you will finally get that promotion tomorrow specifically at 4:30 PM and it happens? It may seem amazing on its face, but considering that you were most likely already in line for that promotion, have been thinking about it a lot, and that many employers make calls like this near the end of the work day, it is not amazing at all. If you also acknowledge that those who believe in the “power” of predictive dreams are actively seeking events in their day to match their dreams and reinforce their belief, we realize that we have purely random coincidences with a pinch of confirmation bias, nothing more. This active seeking combined with randomness is a death knell for many pseudosciences that rely on math illiteracy (i.e. astrology can be rationally explained away in this view).

It boils down to this: unlikely events happen all the time because we live in a random world filled with billions of people. Even when you seemingly predict an event with a dream, we would expect some matches from chance alone. Also, you only predict events that have some prior probability of happening anyway. When you line all of the various probabilities up, a "miraculous" event is bound to happen to someone at sometime. Even one-in-a-trillion events happen to people, it’s just a matter of probability.

If “amazing” coincidences like predictive dreams did not ever happen, we could be suspicious about the workings of our world. However, this is not what we see. We see and expect supremely unlikely coincidences to happen through chance alone. Understanding these facts is the first step to thinking probabilistically and therefore more accurately, inoculating you from some of the pseudosciences that prey on these deficiencies. Overcoming your math anxiety will make you a better critical thinker, and trust me, there’s a good chance of that.

 

Examples from this post were adapted from the book “Innumeracy” by John Allen Paulos.

Kyle Hill is the newly appointed JREF research fellow specializing in communication research and human information processing. He writes daily at the Science-Based Life blog and you can follow him on Twitter here.

Trackback(0)
Comments (19)Add Comment
@kyle hill
written by Davis, March 08, 2012
You said: "if there’s a likelihood that something will happen, however small, it is explained by chance alone that it is bound to happen."

We recently had a friendly "in-house" debate on the JREF site regarding a similar hypothetical. If one billion people all flipped a coin at the same time, would they ever all land on the same side? Our disagreement was this: Since the odds are so remote, it is in reality an impossiblity. However, some believe that odds do not matter at all and that at some point in time all one billion coins would be flipped at the same time with the same side showing. In other words, there is no such thing as an impossible event - because eventually anything can happen if given enough time. I would appreciate your thoughts on this.
report abuse
vote down
vote up
Votes: +0
@Davis
written by SciPhile, March 08, 2012
Hello Davis,

My point was that if there is a small chance of something happening (like having a billion flipped coins landing on the same side) chance alone can explain it happening. However, this depends on the context. For example, if you only flip those billion coins once, there is such a small chance of them all landing on the same side that for practical purposes we can call it impossible (and would consider the actual probability negligible). On the other hand, if you flip those billion coins a trillion times over time, the odds become better (though still remote) and we could not practically call the event impossible.

For the purposes of this article, I mean that things that are one-in-a-billion happen to people all the time because there are billions of people with trillions of life experiences. So I guess it depends on the context you are talking about; if you flipped a billion coins only once, you may call it an impossibility but mathematically there is a chance (however small) that they could all land on the same side. In this context, nothing with a probability of happening is truly impossible. But this says nothing about practicality.

The only things which we could call "impossible" are the things that have exactly zero probability of ever happening (like surviving after having your head cut off and body disintegrated, as a morbid example).

And as to the coin flipping example, consider that all of the coins landing on the same side isn't as crazy as you may think. If you do the math, the chances that a coin flipped five times in a row will land (in terms of Heads or Tails) in the sequence HHTHT or TTHTH are equivalent to the coin landing in the sequence HHHHH or TTTTT.
report abuse
vote down
vote up
Votes: -1
Understanding Coincidence
written by kurtoli2, March 08, 2012
My father was a bit of a mathematician and even considered going back to teaching before his sudden death in 2008. He used to tell me, in regard to the Lottery, not to worry if my numbers came up and I hadn't played them, because based on probability it's just as possible for the same numbers to come up again. Even if it happened on the very next play wouldn't be a miracle. It also doesn't matter if you play a random sequence of numbers, because the chances of 1,2,3,4,5,6 being called are just as probable as any other combination of 6 numbers.
I agree that it isn't practical to consider the 'all coins flipping to the same side' scenario, but it's fascinating to consider that based on probability, when we say that perhaps there's an ever so slight chance it could happen if the coins were flipped repeatedly for millions of years (for example), that there's also nothing to prevent it from happening on the very next toss. This is where the mind wobbles. smilies/smiley.gif


report abuse
vote down
vote up
Votes: +1
Understanding Coincidence
written by kurtoli2, March 08, 2012
Correction: I meant to say, "...there's nothing to prevent it from happening on the very first toss"

Talk about a wobbling mind

report abuse
vote down
vote up
Votes: +1
epends on context
written by sailor, March 08, 2012
Kyle, clearly you are right it depends on context. The chances of any particular hand of cards being dealt is remote, but every time you deal a hand of cards does happen. The relevant bit here is with the particular hand. If you made a prediction that when you dealt a hand of cards you would get the following and then you did you I would be absolutely certain you magician who knew how to handle cards!
report abuse
vote down
vote up
Votes: -1
one billion coins
written by CNS100, March 08, 2012
For Davis: if n people simultaneously flip fair coins, there are 2^n distinct possible outcomes, each equally probable. One will occur, and the others won't. In the case of one billion people flipping, the event "either all H or all T" is twice as probable as whatever actually happens. The lesson is that some very low probability outcomes will indeed be observed.

Better yet, consider a continuous probability distribution -- e.g. random selection of a point on the real number line. Regardless of what is selected, the probability of selection was zero. Zero probability events aren't the same as impossible events; a fortiori neither are low probability ones.

Always useful to keep in mind if you hear a creationist try to calculate (OK, make up) the probability the universe "could have happened accidentally," and pronounce it too small to be believed. Such exercises are fundamentally confused.

report abuse
vote down
vote up
Votes: +0
...
written by lytrigian, March 08, 2012
You never see a great white shark with braces or are suddenly chased down the street by a cheeseburger.

OK, does anyone ever actually have dreams like that? I'm sure I never have. It sounds like the kind of thing someone blurts out for a cheap laugh in a sitcom.

I think in the billion coin flip thought experiment, we can safely assume that any one outcome we attempt to predict will certainly not happen. Bear in mind the sheer magnitudes here. Granted that "all T OR all H" is twice as probable as any single outcome, that only lowers the probability from 2^(10^9) to 2^((10^9)-1). This is a staggeringly huge number. You have a far, far better chance of finding a molecule of the "active" ingredient in a 30C potency homeopathic remedy than of predicting the result of such a coin toss. But this is even true of a much smaller (and, incidentally, quite doable) experiment of only 1,000 simultaneous coin flips. The number of possibilities there is at least easily computable with an ordinary calculator: 2^1000 = 1.07 x 10^301 possible outcomes.

Better yet, double the sample size to 2000. You'd then be within an order of magnitude of a 300C homeopathic dilution, which is what they use for the popular oscillococcinum remedy. (about 10^602 vs 10^600) It would make for a dandy illustration of the situation.

In any case the number of possible outcomes outstrips by many orders of magnitude the total number of protons in the universe. I'd question whether enough time even exists, in an absolute sense, to carry it out often enough to guarantee any given outcome with even 1,000 coin tosses, let alone 1,000,000,000. Predicting such an outcome by claimed psychic means (precognitive dreams or otherwise) would come as close as anything possibly could to absolute proof of the paranormal. It would indeed make it more certain than a great many other things we accept as true.
report abuse
vote down
vote up
Votes: +0
...
written by lytrigian, March 08, 2012
Granted that "all T OR all H" is twice as probable as any single outcome, that only lowers the probability from 2^(10^9) to 2^((10^9)-1). This is a staggeringly huge number. You have a far, far better chance of finding a molecule of the "active" ingredient in a 30C potency homeopathic remedy than of predicting the result of such a coin toss.

The sad result of confused editing. I had meant the 30C comparison to relate to a 1,000 coin experiment. In either case the homeopathic dilution pales: a mere 10^60 vs. 10^301 for the thousand coins. Next to these numbers, 2^(10^9) isn't even worth considering: subtracting them both out would be lost among the insignificant figures.
report abuse
vote down
vote up
Votes: +0
Random thoughts....
written by CatOfGrey, March 08, 2012
With dreaming predicting life events, you can't fully determine whether the 'aha' moment is actually remembering that a current situation was foretold in a dream. It may be deja vu, or some cognitive bias that triggers the thought.

As for the mathematics of coincidence, I prefer my catch phrase: "The world has 7 billion people in it. That means that 7000 people each day have a 'one-in-a-million' kind of day." Some will win the lottery, and some will crash their car, go to the hospital and discover they have cancer. The irony of it all is that 'rare' events are surprisingly common.

report abuse
vote down
vote up
Votes: +0
...
written by Willy K, March 09, 2012
Kyle, while I agree that math education is a vital tool in a Humans mental tool kit, I don't believe it is the major problem and/or solution to people not understanding coincidence or statistics when applied to themselves.

I say this because of my non-scientific anecdotal observations of two groups of people. smilies/tongue.gif

Smokers and gamblers.

I know many smokers, both high IQ and low IQ. They have before all the statistics and evidence they need to make a sound decision to stop smoking. The most common thing I've heard them say is "But my uncle (typically some relative of theirs) smoked two packs a day until he was 90!" They simply reject all negative information because they believe that statistics do not apply to them.

Gamblers I've met always have a "system" to beat the odds. They believe their attitude will affect the outcome of games of chance. They never admit to losing, they always brag about their winning!

I should
report abuse
vote down
vote up
Votes: +1
...
written by Willy K, March 09, 2012
Oops, "I should" stop typing now! smilies/cheesy.gif
report abuse
vote down
vote up
Votes: +0
...
written by Mark P, March 09, 2012
Comprehending this deficit and doing something about it should be taken up within our school system; we should engage students with math early, often, and more rigorously.


We do. Math is compulsory through the whole of school.

Moving complex ideas earlier will not work. Kid's brains are not mature enough to cope. Sure, you can do what the Asians do and get them rote learning, so that on the face of it they are better at math. I've taught Asian educated students alongside Western educated ones, and that rote learning does not give them any more understanding. Is rote learning actually what you want?

Obviously we should teach Math better. We should also give every person in the world enough to eat and freedom to chose their leaders. We don't because of large – very nearly impossibly large – difficulties. It's not like their is a country in the world where the politicians say "Hey, let's skimp on basic math and reading".
report abuse
vote down
vote up
Votes: -2
...
written by Caller X, March 09, 2012
if there’s a likelihood that something will happen, however small, it is explained by chance alone that it is bound to happen to some people at some time.


No, it's explained by chance alone that it might happen. Otherwise, cool story bro.

Similarly with the coin flipping. You could flip a billion coins forever and never have them come up all heads or all tails.
report abuse
vote down
vote up
Votes: +0
Monty Hall Dilemma
written by Zoroaster, March 09, 2012
One of the coolest 'Mythbusters' episodes I've seen is the one that demonstrated the 'Monty Hall Dilemma'.

On Let's Make a Deal the contestant would be given the choice of three doors, the prize could only be behind one of them. The host knew where the prize was. The game began with the contestant choosing their door. To add suspense the host would open one of the doors they had not chosen to reveal it was not where the prize was. This left two doors, one with the prize, one without. The player had supposedly already chosen their door but now Monty offers them the opportunity to change their mind and switch to the other remaining door. On Let's Make a Deal, most contestants stuck with their choice, figuring that their intuition was all they had to go on and they had a 50% chance of being right. What Adam and Jamie showed was that there was actually a distinct advantage to switching. The mathematics was shown and then they played the game over and over and sure enough, switching increased ones odds from one in three to one in two. It still confuses me to think about it though.
report abuse
vote down
vote up
Votes: +2
...
written by Caller X, March 09, 2012
One of the coolest 'Mythbusters' episodes I've seen is the one that demonstrated the 'Monty Hall Dilemma'.


I thought the one where they made the suit of armor out of paper was equally good, and it had exactly the same connection with coincidence: none.
report abuse
vote down
vote up
Votes: +0
Lottery numbers
written by EarlyOut, March 10, 2012
@Understanding Coincidence: The problem with playing 1, 2, 3, 4, 5, 6 in a lottery is that almost all lotteries are parimutuels. In other words, if you're the only person who gets the right numbers, you get the whole jackpot. But if other people also hit it, you have to split the prize. Playing obvious sequences (1 through 6, or 2, 4, 6, 8, 10, 12) increases the odds that if your numbers come up, someone else will have played the same numbers. You're slightly better off with numbers that have no clear pattern.

Of course, the chances of winning are so small to begin with, you might as well just wish for $100 million. At least wishing won't cost you a buck!
report abuse
vote down
vote up
Votes: +0
legitimate ?
written by rosie, March 10, 2012
I agree that many people have a fear of numbers, equations, and probability, but why do you call it "legitimate"? I think it something the whole country should be ashamed of.
report abuse
vote down
vote up
Votes: +0
...
written by MorganD, March 10, 2012
This type of "chance" is true not only in the lottery, but in slot machines as well. There is no reason to believe that if someone hits the jackpot and leaves the machine that another jackpot won't hit just a few minutes later or even with the very next spin. It takes a lot to understand probability and math and most people shy away from what they don't understand.
 

 

casino en ligne

Gambling at an casino en ligne is a great way to win casino cash and playing with a casino bonus is even better.

report abuse
vote down
vote up
Votes: +1
@Zoroaster re;Monty Hall problem
written by tmac57, March 11, 2012
Zoroaster-
switching increased ones odds from one in three to one in two.

I think you meant that it increased their odds from 1 in 3 to 2 in 3.
report abuse
vote down
vote up
Votes: +2

Write comment
This content has been locked. You can no longer post any comment.
You must be logged in to post a comment. Please register if you do not have an account yet.

busy