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.
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):
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