The patterns of probability produce some unusually interesting alignments. Here's how to interpret coincidences that seem unbelievable.
One of the occasional sad duties of statisticians is to take a world full of whimsy, delightful serendipity, and surprises around every corner and turn it into a dull, predictable, uninteresting place. I'm about to do that here, so if you would rather keep wearing rose-colored glasses, put them on now, skip this hack, and pick another one (I suggest more pleasant topics, such as winning Monopoly [Hack #51]).
I choose to be scientific and treat the world as rational and built on consequences that follow chains of cause and effect. My problemand perhaps yours too, if you think like meis that when I face anomalies (hard to explain, unexpected things), it is tempting for me to treat the happening as evidence of something mystical, or psychic, or paranormal in some way. Coincidences are a good example. When I witness an incredible coincidence, I am tempted to fall into a comforting pit of nonscientific explanations, such as fate or synchronicity.
The solution to my problemand perhaps yours, if you're still with meis to think a bit and apply some basic rules of probability. This way, I can get a handle on things and treat such coincidences as inevitable considering the large sample sizes that exist in the universe. By applying such rules, I can feel better about the world I live in. I can sleep peacefully in the arms of chance, and I have no need for mystical, magical explanations. Here are three strategies for tackling the next amazing coincidence you come across.
Compare the Number of Possible Outcomes
When I was a kid, I used to see a common advertisement in the comic books I read (e.g., Statboy and His Flying Dog, Parameter). The ad sold U.S. pennies that had been altered to include a portrait of John F. Kennedy in addition to the standard Lincoln profile. To justify why these two presidents should be included together, a long list of "remarkable" coincidences shared by these two presidents was presented (and, as I recall, if I purchased a set of these pennies, I would even get a small poster that listed these similarities.)
The list included things beyond the obvious, such as the facts that both were assassinated and both were succeeded by vice presidents named Johnson. I could (and did) interpret these coincidences as evidence of some important, somewhat-magical connection between the two. Let's use these coincidences as an example and approach it as a research question: is there an unusual number of similarities between these two presidents?
One tool to use when deciding whether a coincidence is remarkable or predictable is to count the number of possible outcomes and then determine whether the given outcome (the coincidence) is unlikely to have occurred by chance. This is the approach taken when predicting shared birthdays in a large group [Hack #45].
Column one of Table 6-1 presents a list of some of the coincidences shown in those old comic book ads and also found in "Hard to Believe"-type publications. Column two shows a brief list of characteristics that both men could have shared, but did not.
By paying attention to only the relatively few concordances between Lincoln and Kennedy (the hits) and ignoring all the non-hits, of which there are almost infinitely more, it is easy to misperceive the existence of some uncanny link. Of course, there still might be some uncanny link, but the "coincidences" do not provide evidence for it.
Figure Out the Actual Odds
If you play poker with any regularity (and, if you are a minor Hollywood celebrity, you apparently play all the time), you know that you rarely see a royal flush: a five-card hand with the 10, Jack, Queen, King, and Ace all of one suit. If your opponent were dealt a royal flush, would that be remarkable? Would you suspect cheating? It all depends on how many poker hands you have seen in your lifetime, I guess, or perhaps in recent memory.
Let's use a simple deal of five cards to do our math. To figure the chances of getting a royal flush on one deal of five cards, we would first calculate the number of possible five-card poker hands and compare that to the number of those combinations that are defined as a royal flush. The process takes three steps:
Your opponent or you should be dealt five cards that make a royal flush once every 649,740 hands. So, if it does happen, it is certainly rare. If it happens more than once in the same game, you should interpret that as an amazing coincidence or as evidence of cheating. You decide. I know what my calculator and I would guess.
Remove Meaning Assigned to Meaningless Events
The human brain is at its best when it must make meaning out of data. Our remarkable intelligence can find meaning even where there is none. Often, this is the case when we think we have witnessed a miraculous set of coincidences. We see coincidences when we look for them.
Highly improbable events happen all the timeevery day, and every minute of every hour. The highly improbable events are interesting only when we decide they are interesting. Think of our poker example. Because there are about 2.6 million possible five-card poker hands, the chances of any specific hand are one out of about 2.6 million. The odds are the same for the hands we have decided are particularly meaningful, such as a 10, Jack, Queen, King, and Ace of Spades, as they are for hands that we have decided are not particularly meaningful, such as a 4 of Clubs, 6 of Spades, Jack of Diamonds, Queen of Spades, and Ace of Hearts. Why is it amazing that you just drew a royal flush and not equally amazing when you draw any other random combination of cards? The probability is the same for all poker hands. We assign the meaning to a particular outcome.
The next time you are at a crowded place, such as a baseball game, amusement park, or airport, and you run into someone you know, notice that the coincidence is meaningful only because you happen to know the person. Yes, the chances were slim that you would run into that particular person (unless you are being stalked), but it is 100 percent certain that you would run into other people. All those other people just happen to be there the same time you are. It is a coincidence, and it is highly improbable that this particular mix of individuals is in the same place at the same time. It is not a meaningful coincidence for you, though.
We are constantly exposed to a large set of events and people and things that interact and coincide in very unlikely ways. Occasionally, those coincidences have meaning to us, and so we notice them. What is amazing is that we do not notice these highly improbable events more often.