The leg starts. A player approaches the dartboard, cool as ice. With a fluid motion, he bends his elbow, leans in, and releases a dart that finds its mark in the triple 20 as if it were the most natural feat in the world. Within a second, he follows with another throw, and another.  Again, a perfect throw. Then another, and another, and another. By the time he reaches his seventh dart, the arena has fallen into silence. Everyone holds their breath and waits in anticipation. Two darts are left and 81 remains to be thrown. Triple 19. Just one dart left. Double 12. The place explodes. Nine darts, zero mistakes, and most importantly, game over. Surely a player this good could be doing this all the time? That, however, is completely false. 

Finishing a leg in nine darts is not just great, it is the minimum number of darts you can throw to win a leg of 501. The game is as follows. Players take turns throwing three darts and the points thrown are deducted from the initial amount of 501. One wins the game by getting the amount down to exactly 0 with the final dart being a double or bull. There are many ways to achieve a nine-darter. In fact, there are as many as 3944. However, since a triple 20 scores the highest, the most common path is seven triple 20’s in a row, leaving 81. This can then be cleaned up by a triple 19 and a double 12. Despite decades of professional competitions, only a handful of nine-darters have ever been broadcast on live television, whilst there have been thousands of legs played.

The mathematics of consecutive probabilities explains this phenomenon. Our intuition is terrible at understanding how difficult it actually is to repeat something correctly many times in a row. We tend to think of a 50% chance as reasonably high, high enough to feel comfortable. However, the numbers turn against you very fast if you want to succeed repeatedly with that 50% without a single failure. Imagine that you want to throw a fair coin 10 times with heads landing on top every throw. Even though each flip is a 50/50 chance of getting a head, if you want to have 10 in a row, then the probability is 0.5 to the 10. Hence, this will actually take you more than a thousand tries on average.  

The product rule of probabilities does not just appear in statistical experiments. This rule appears everywhere. Imagine an incredible surgeon who can do each step of a procedure with 99% accuracy. That sounds like a relatively safe surgery right? What if, however, the surgery contained 50 independent steps? In that case, the surgery is only completed successfully around 60% of the time, which is 0.99 to the power of 50. So, even the best doctors can fail difficult surgeries regularly, this is also a depressing real life fact that comes from probability. This hopefully draws a distinct line between near-perfection and true perfection. You can see this effect everywhere. A pilot running through a 30 step checklist without a single oversight. A driver navigating a three-hour route in a snowstorm without missing a turn once. A teacher getting through a whole parent-teacher night without accidentally mixing up two kids' names. The more steps required, the more difficult it is to succeed. 

Getting back to darts, imagine a professional darts athlete who can hit any smaller segment (double or triple) 50% of the time. This would already make him one of the very best in the world. In order to calculate the probability of a nine-darter with the product rule, we need independence. Unfortunately, throwing multiple darts in a row towards the same spot can increase your accuracy, making the throws dependent. On the other hand, shooting towards the same point can also increase the chance of a dart getting knocked out by another. For the sake of simplicity, I will assume that these effects cancel each other out and will naively assume independence. The probability of throwing a nine-darter, is hence 0.5 to the power of 9, which is 0.2% of the time. This translates to roughly one nine-darter every 500 legs. An athlete that plays darts full-time, throws a few hundred legs in a season and hence such a spectacle is almost never seen in a regular tournament. 

To put that into perspective, consider what happened when Van Gerwen came closer than almost anyone ever has. He once hit 17 consecutive perfect darts. Just one shy of two perfect nine-darters in a row. The chance of that happening is 0.5 to the power of 17, roughly one in 131,000. It is the kind of sequence that may genuinely never be seen again in our lifetimes. Something so statistically improbable that luck must have played a role, yet so precise that you cannot possibly give chance all the credit either. Now, consider what those numbers mean for an amateur. A decent player might hit the required segment on around 10% of throws. For them, the probability of a nine-darter is 0.1 to the power of nine, which is one in a billion. If that amateur threw nine darts every minute without stopping, they would need to keep throwing for roughly 19,000 years before expecting their first perfect leg. That is longer than all of recorded human history. The nine-darter is not merely difficult for an amateur. It is practically impossible.

We often find ourselves frustrated when elite athletes fall short of perfection. A missed penalty in the final minute, a serve that clips the net at match point, a putt that slips out on the 18th hole. Our instinct is disappointment, as though the professional has let us down by being human. However, it is precisely this randomness, this uncertainty that no amount of practice can fully master, that makes sport worth watching in the first place. If the best player always won, if the nine-darter was routine, if the outcome was determined entirely by skill, there would be nothing left to cheer for. The beauty of sport lives in the gap between what athletes can almost do and what they occasionally, miraculously, actually do. The nine-darter is not a reminder that professionals are imperfect. It is a reminder that perfection, when it appears, is genuinely extraordinary.