I found Scott Sumner’s recent exploration of our misconceptions regarding coincidences both enlightening and amusing. Our tendency to misinterpret random events stems from a variety of factors, but one key reason is our selective acknowledgment of certain coincidences, which makes them seem far more extraordinary than they truly are.
Take, for instance, a personal anecdote from my childhood. My family had a tradition of playing spades, typically with my dad and me teaming up against my mom and younger sister. Occasionally, I would find myself dealt a hand that prompted the thought, “What are the odds of receiving a hand like this?” However, I often caught myself mid-thought and reminded myself that the probability of receiving that specific combination of 13 cards was no different from any other combination.
Why did I react this way to certain hands but not others? My knee-jerk response typically arose when I received a hand that was strikingly unusual, particularly one that would greatly influence my potential success in the game. If I happened to receive seven spades, my hand would be exceptionally strong, leading me to anticipate a higher number of tricks. Conversely, if my hand consisted exclusively of cards valued seven or lower, I would be inclined to consider a zero bid. Yet, most hands I was dealt appeared fairly average, with a balanced assortment of black and red cards, suits, and values.
To illustrate this with an extreme example, consider the following two spades hands:
- Hand one: Ace of Spades, Seven of Hearts, King of Clubs, Two of Diamonds, Ten of Spades, Five of Clubs, Jack of Hearts, Three of Spades, Queen of Diamonds, Nine of Spades, Six of Hearts, Eight of Clubs, Four of Spades.
- Hand two: Two of Spades, Three of Spades, Four of Spades, Five of Spades, Six of Spades, Seven of Spades, Eight of Spades, Nine of Spades, Ten of Spades, Jack of Spades, Queen of Spades, King of Spades, Ace of Spades.
With the first hand, I would casually assess my options for bidding but would likely not dwell on it further. However, if I were dealt the second hand, I would be utterly astonished, convinced this was a rare and miraculous occurrence that no one would believe. In fact, if I were playing against someone who received that second hand, I’d probably suspect foul play (or perhaps that they were a skilled magician, which amounts to the same thing).
Despite both hands being equally probable, the second hand feels far less likely because it deviates from our conventional expectations of randomness. The first hand represents a typical strength, while the second hand is unbeatable. This discrepancy is why I would overlook the extraordinary nature of the first hand. Even though the odds of obtaining that first hand are astronomically low (approximately 1 in 635 billion*), its impact is not immediately apparent. Each time you receive a 13-card hand in spades, you are witnessing an event that is thousands of times less likely than winning the Powerball—yet we tend to disregard such coincidences.
(*The total number of possible 13-card hands is calculated as 52! / (13! * (52 – 13)!), yielding 635,013,559,600 unique hands.)
If I were to tell you, “The odds of X happening to you are about 1 in 635 billion,” you might reasonably conclude that such an event is unlikely to occur in your lifetime. Yet, every time you are dealt a hand in spades, you experience a 1 in 635 billion event. These massively improbable occurrences happen regularly, but we rarely take notice.
Even more astonishing is the arrangement of a shuffled deck of cards. There are 52! different ways to arrange a deck—an unfathomable number represented as:
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
This mind-boggling figure illustrates that every time you shuffle a deck, you are likely creating a configuration that has never existed before and will never exist again, barring the heat death of the universe. Yet, even with this knowledge, I rarely find myself awestruck by the improbable odds of the arrangement I just created—unless it exhibits some remarkable characteristic (perhaps an alternating pattern of red and black cards).
Although I understand this intellectually, I still struggle to internalize it instinctively. Hence, my initial reaction to certain spades hands feels disproportionately improbable. While my “system one” mind may react impulsively, it’s crucial to engage my “system two” thinking to remind myself that the mundane occurrences around us are just as astonishingly unlikely as those that seem more extraordinary. That dramatic coincidence that initially captivated my attention may not be as exceptional as it appears.
