Last month an article on one of my favorite websites, io9.com, grabbed my attention. It included a discussion of studies and simulations which demonstrate (and provide evidence for) some of those things in life that lead us all to think that fate is trying to tell us something. Specifically, the adage we call “Murphy’s Law” states that what can go wrong will go wrong and it is supported by both mathematical proofs and observations.

“When it comes to long strands of string, from proteins in a person’s cells to the rigging in a ship, this means spontaneous knotting. People have written papers about how string knots up the minute it’s given a chance to jiggle around.”

The article goes on to discuss a simulation of a random walk (direction for each step is determined randomly) in 3-dimensional space with the restriction that no space can be occupied more than once. The path of the walk simulates the placement of a length of string – the beginning and end of the path are the ends of the string and no part of the string can occupy the same space as another part. What the researchers found was that any sufficiently long walk (string) must contain a knot. The longer the walk (string), the more knots.

This can teach us that tossing our Christmas lights into a box is almost certain to result in knots to untangle next year, but it can also teach us a lot about risks, coincidences, and how to think about those things.

Barbara Drescher

When I was nine years old my family lived on the Great Lakes Naval Station (on the shores of Lake Michigan) for about a year before buying a house off the base. Our home was in a cul de sac that was shared by several families. One of the families of which we were particularly fond was a widower with a boy and girl about the same ages as my brother and I. Fast forward to more than four years later, after we had moved twice and now lived 2,000 miles away in Sacramento, California. We drove the three hours from our home to a tiny fishing hole called Blue Lake for a weekend of camping and fishing. About an hour after we arrived, my mother suddenly blurted out, “Hey, isn’t that Bud Neighbor [not his real name]?” Sure enough, camped a few spaces down were our old friends.

I have had quite a few similar experiences, but none as bizarre or unexplainable as this one. Should we have been freaked out and considered some cosmic connection?

To find out, let’s turn back to the article I mentioned and Murphy’s Law. Is it true that anything that can go wrong will go wrong? Well, not exactly. You can get on that plane tomorrow and be confident that you will survive the flight (your odds are approximately 9.2 million to 1). However, if you “tempt fate” enough, even the least likely disaster will eventually happen. Of course, your plan to commit suicide via commercial airliner will require you to fly every day for more than 25,000 days to ensure success, and even then you have no guarantees.

The problem of spontaneous knotting is simply a matter of odds. It relies on something called “The Law of Large Numbers”, which dictates that any event which can occur will occur if given enough opportunities.* String knots up when there’s a lot of it because there are a number of ways in which it can be knotted.

Without going through a bunch of math, let’s look at how we determine probabilities. There are two properties to consider when determining whether an event is unusual (vs. expected):

1. How likely is it to occur in a given instance (a trial)?
2. How many chances (trials) has it had to occur?

For example, if we want to know the probability of rolling a “six” given a single die, we must know the odds of rolling a six on a single roll (a trial) AND how many times the die is rolled (number of trials in a sample). If we roll once, the odds of a six are one in six (1:6).  If we roll it twice, then the odds of at least one of those rolls being a six is 2:6 or 1:3.  A sample of three rolls brings the odds of at least one six to 1:2, or 50/50.

What about getting two sixes in a row? This is more complicated, but only slightly. In two rolls, the odds of both being sixes are 1:36. If, however, we roll the die ten times, the odds of at least two consecutive sixes is much higher (1:4) because there are nine chances to get two in a row (on rolls 1 & 2, rolls 2 & 3, and so on). So, even if the probability of getting ten sixes in a row is small (in one trial of 10 rolls, it’s less than 1 in 20 million), if we do it enough times it will happen. For a great demonstration of this, watch Derren Brown flip a coin and get heads ten times in a row (the odds of which are less than 1 in 1,000), then watch the full show (available on the same YouTube channel in six parts) in which they reveal the “trick”.

It is easy, however, to misinterpret the Law of Large Numbers and Law of Averages, confusing the odds on a given, independent trial (the roll of one die) with the long-term odds of a set of trials. This misinterpretation is often called “Gambler’s Fallacy” because gamblers are particularly vulnerable to it. For example, would you sit down to a slot machine after someone has just hit the jackpot? Most people would not. Would you play one that has been losing for hours? Many would, thinking that it is “due to hit”. The truth is that the likelihood of hitting a jackpot is the same each time you pull the handle, however, if you calculate the odds of hitting it at least once in a set of one hundred pulls, your odds are much better – one hundred times better, to be exact. It doesn’t matter what happened before you sat down, nor does it matter how many times you have already pulled the handle, but it does matter how many times you plan to pull the handle before giving up.

What the Law tells us is that if you sit long enough at the machine and put in enough coins, you are virtually certain to win. But randomness is cruel; you don’t know if it will happen on the first pull, the tenth pull, the hundredth pull, or the millionth pull.

So what does this have to do with running into our former neighbors at a tiny lake in northern California four years after living near them two thousand miles to the east? It’s a highly unlikely event by itself, but when we consider how many opportunities exist for highly unlikely events to occur, the probability tables are turned.

It’s not easy to quantify the likelihood of such things because there are a number of variables (e.g., how likely were both families to choose camping that weekend?), but let’s try a ballpark estimate. Consider the two things I mentioned:

1. How likely is the event?
2. How many chances has it had to occur?

If we define “the event” as running into that particular family, number 1 is extremely unlikely, but we must also factor in some post-hoc thinking. The truth is that it wasn’t running into them that was shocking, but seeing anyone that we knew that we didn’t expect to see (let’s say there were only 200 people in the country who would fit that description). The odds of being surprised to recognize at least one of the 100 people at the campground in a nation of about 225 million (at the time) are a bit less than 1 in 100,000 – not as low as you might have expected, but still pretty low.

Here’s where the random walk simulation comes in: even though the odds are low, we have to multiply that by the number of opportunities for it to happen. We run into people everywhere – the supermarket, the mall, while bowling, at ball games and concerts, meetings, on the street, getting coffee, etc. For our purposes, let’s limit our estimate to the odds of this happening on a weekend. In a one-year period alone, there are 52 opportunities, so our odds were better than 1 in 250. Over ten years there were 520 “trials”, so the odds were nearly 1 in 20. Over a lifetime, it would be highly unusual NOT to have at least one bizarre experience like the one I described.

In a nutshell, what are the odds that you will run into someone totally unexpected at some point in our life?  Well, the odds are excellent that it will happen at least a few times, for the same reason that the odds are you’ll have more than one knot in your Christmas lights next year.

*There are variations of the Law of Large Numbers and it should be noted that most address the central tendency or expected value on a single trial. However, inherent in this law, particularly in Borel’s law, is the property that, in large samples, the proportion of occurrences of any given event matches its probability so that any event with a non-zero probability will occur if the sample is large enough.

Barbara Drescher is a former educator and researcher, having taught research methods, statistics, and cognitive psychology at CSU Northridge for a decade. At ICBSEverywhere.com, Barbara evaluates claims and studies, discusses education, and promotes science and skepticism.