Hack 62. Demystify Amazing Coincidences

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.

Synchronicity is the term that pioneering psychiatrist Carl Jung used for personally meaningful coincidences. He saw them as providing insight into the inner world of the unconscious, but was not above assigning pseudo-mystical explanations to them as well. He was not a statistician.

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?

It occurs to me now that the comic book ad led me to think for a time that the word coincidence was derived from the word coin. Of course, I quickly learned otherwise (by graduate school, certainly) that it has something to do with co-incidents.

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.

Table Comparing Abraham Lincoln and John F. Kennedy

Some amazing coincidences	Some unremarkable noncoincidences
Both assassinated.	Different heights.
Both elected in years ending with 60.	Different weights.
Kennedy assassin shot from a warehouse and hid in a theater. Lincoln assassin shot in a theatre and hid in a warehouse (well, a barn anyway).	They were different ages when they died (though they were the same age when they were born).
Lincoln was shot in Ford's theater. Kennedy was shot in a Ford.	They were born on different dates in different years.
Both were killed on a Friday.	Both men had different middle names.
Both were killed while sitting next to their wives.	Both men had wives with different names and, probably, different shoe sizes.
Both succeeded by men named Johnson.	Succeeded men with different names.
	Lincoln had a beard; Kennedy did not. (Come to think of it, their faces are different in hundreds of ways.)
	Kennedy probably had bowled occasionally; Lincoln never bowled a game in his life.

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:

1. Calculate the number of possible hands, if order makes a difference. We start this way because the math is easiest. Any one card of 52 could be the first card, then any one of the remaining 51 could be next, then any one card out of 50, and so on down to any one card out of 48. So, the number of possible hands when the order matters is:

1. Order does not matter, though. So, we divide this giant total of all possible hands by the number of possible different sequences of cards. This number of different sequences is 5x4x3x2x1 = 120, so the number of possible five-card poker hands is:

1. Because there are only four possible royal flushes, one for each suit, we divide this number of positive outcomes (4) by the total number of possible outcomes (2,598,960), for a probability of .000001539, or 1 out of 649,740.

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.

What about drawing to a royal flush? After all, in draw poker and in Texas Hold 'Em, players have an opportunity to improve their hand or at least guide it toward some objective. In draw poker, if you have four cards to a royal flush and wish to discard the fifth and draw a new card, you have a 1 out of 47 chance for success, or .021 percent. If you have two chances to improve your hand, the odds go up to .043 percent, or about 1 out of every 25 attempts.

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.

The odds are even good that you would run into someone you know, if we count anybody you know. Let's say you know 200 people and you, by yourself, go to a Kansas City Royals baseball game one night. If each of those 200 people goes to a Royals game one time each season and there are 81 home games, each of those 200 people has a 1/81 chance of being there the same night as you. It's unlikely then that you would run into any particular personsuch as your Uncle Frank, for examplebut it is highly likely that someone you know will be there. There is about a 92 percent chance that one or more of your 200 pals will be there, even though each of them rarely goes to a game. Even if you only know 56 people, the chances are greater than 50 percent that one or more of them will be there.

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.