The Mathematics of Darts: Probability, Geometry, and Optimal Play

Darts looks simple. Throw pointed objects at a numbered board, subtract the score, hit a double to finish. But beneath this simplicity lies a rich mathematical structure. The dartboard is a masterpiece of applied geometry, designed to punish inaccuracy in ways that are not immediately obvious. The optimal place to aim depends not on the highest number, but on your personal accuracy. And the seemingly straightforward act of finishing a leg involves probability calculations that separate great players from good ones.

The Geometry of the Dartboard

A standard dartboard is 17.75 inches (451 mm) in diameter, divided into 20 numbered segments by thin metal wires running from the outer edge to the bullseye. Each segment contains four scoring zones: the large single area (closest to the center), the treble ring (a narrow band approximately 8 mm wide), the outer single area, and the double ring (another narrow band at the very edge, also approximately 8 mm wide). At the center sit the single bull (a ring with a 15.9 mm inner diameter) and the double bull or bullseye (8 mm diameter).

The numbers are arranged in a specific order around the board, reading clockwise from the top: 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5. This sequence is universally credited to an optimization designed to punish erratic throwing, though whether it is truly optimal in a mathematical sense has been the subject of academic debate.

Why the Numbers Are Arranged This Way

The key design principle is to place high numbers adjacent to low numbers. The 20 sits between 1 and 5. The 19 is flanked by 3 and 7. The 18 is between 1 and 4. If you aim for a high-scoring segment and miss to either side, you land on a low-scoring segment.

Mathematicians have analyzed whether the standard layout truly minimizes the expected score for inaccurate throwers. In 2019, a study by researchers at the University of Manchester used computational optimization to explore all possible arrangements of the numbers 1 through 20 around a dartboard. They found that the standard layout is very good but not mathematically perfect. An optimal layout could reduce the expected score for a random thrower by a small amount. However, the standard layout is within a few percent of optimal, and its 130-year history means it will never change.

What makes the arrangement clever is that it penalizes inaccuracy in both directions. A player aiming at treble 20 who misses high or low lands in the single 20 (still decent), but a player who misses left or right lands in 1 or 5 (terrible). This creates a meaningful risk-reward trade-off: the highest-scoring area of the board is also one of the most punishing if you miss laterally.

Where Should You Aim? The Expected Value Problem

Ask any casual darts player where they aim, and the answer is almost always "treble 20." After all, treble 20 scores 60 points, the highest single-dart score on the board. But for many players, treble 20 is not the mathematically optimal target. The reason is that expected value—the average score you will achieve over many throws—depends on your accuracy, not just the maximum possible score.

Modeling Dart Throws as a Gaussian Distribution

A dart throw can be modeled as a two-dimensional Gaussian (normal) distribution centered on the aiming point. The throw has some horizontal variance (left-right scatter) and some vertical variance (up-down scatter). For most players, these are roughly equal, producing a roughly circular scatter pattern. A professional player might have a standard deviation of 10–15 mm. A good club player might have a standard deviation of 20–30 mm. A casual player might be at 40–50 mm or more.

By overlaying this probability distribution on the dartboard geometry, you can calculate the expected score for any aiming point. The result is an "expected value map" of the entire board, and it changes dramatically with skill level.

Professional Players: Aim at Treble 20

For a player with a standard deviation of about 15 mm or less, the optimal aiming point is at or very near the center of the treble 20 segment. Their scatter pattern is tight enough that the majority of darts land in the treble 20, with most misses going into the large single 20 above or below. The expected value is around 45–55 points per dart, depending on the exact precision.

This is why professional players overwhelmingly aim at treble 20. With their level of accuracy, the risk of landing in 1 or 5 is low enough that the high reward of treble 20 (and the decent consolation of single 20) makes it the clear choice.

Club Players: The Case for Treble 19

For a player with a standard deviation of 25–35 mm, something interesting happens. The expected value of aiming at treble 20 drops because a significant proportion of darts miss into the 1 and 5 segments. Meanwhile, the treble 19 segment is flanked by 3 and 7—which are low, but not as punishing as the 1 adjacent to 20. More importantly, the single 19 area is generous, and misses into 3 and 7 still yield reasonable scores.

Several academic studies have confirmed that for players with moderate accuracy, aiming at treble 19 yields a higher expected value than treble 20. The difference is small—perhaps 1–3 points per dart on average—but over the course of a long match, this adds up. A player who throws 60 darts in a match gains 60 to 180 points by making the optimal choice.

Beginners: Aim for the Center

For a player with very low accuracy (standard deviation above 50 mm), the optimal aiming point moves toward the center of the board, near the bullseye. This is counterintuitive, since the bullseye is a small target. But the point is not to hit the bullseye—it is that aiming at the center minimizes the chance of hitting the very low-scoring segments at the edges. The inner single areas around the bull are all reasonably scored (the numbers 1–20 are all represented), and the expected value is more evenly distributed.

In essence, as accuracy decreases, the optimal strategy shifts from "aim for the highest target" to "aim for the area where bad misses are least likely."

The Mathematics of Checkout

The endgame in 501 is where mathematics becomes most directly useful. When a player's remaining score drops to 170 or below, the game becomes a checkout problem: find a combination of up to 3 darts that reduces the score to exactly zero, with the final dart landing in a double.

How Many Checkouts Are Possible?

The highest possible checkout is 170 (T20, T20, D-Bull). The lowest is 2 (D1). Between these extremes, there are 161 achievable checkout scores. The remaining scores—169, 168, 166, 165, 163, 162, 161, and 159—are called "bogey numbers" because they cannot be finished in 3 darts with a double.

Why is 169 impossible? The maximum score from two darts is 120 (two treble 20s), leaving 49. But 49 is odd, and all doubles are even, so you cannot finish on a double from 49 with one dart. You would need to throw a single to make it even, but then you have already used three darts. Similar arithmetic eliminates the other bogey numbers.

Why D16 Has the Highest Success Rate

Professional statistics consistently show that double 16 has the highest success rate among all doubles on the board. This is not because D16 is physically larger (all doubles are the same width) or inherently easier to hit. It is because of what happens when you miss.

If you aim at D16 and miss inside (hitting S16), you are left with 16—which is D8. Miss D8 inside, and you are left with 8—which is D4. Miss D4 inside, and you are left with 4—which is D2. Miss D2 inside, and you are left with 2—which is D1. Each miss leaves you on another double. This is called a "doubling down" path, and D16 provides the longest such path on the board: D16 → D8 → D4 → D2 → D1.

Compare this to D20. If you miss D20 inside, you are left with 20, and you need S4, D8 (two darts). The recovery path is shorter and less forgiving. This is why professional checkout tables overwhelmingly route players to D16 when there is a choice. A player who averages 40% on doubles will check out from 32 (D16) significantly more often than from 40 (D20) over a large sample of attempts, purely because of the favorable miss path.

Checkout Strategy: Route Selection

When a player has a remaining score that can be finished in multiple ways, the choice of checkout route is a probability problem. Consider a remaining score of 76. Two common routes are:

• Route A: T20 (60) leaves 16 → D8. If T20 misses into S20, left with 56 → T16, D4 (recovery possible).
• Route B: T16 (48) leaves 28 → D14. If T16 misses into S16, left with 60 → S20, D20 (recovery possible).

The professional checkout table recommends T20, D8 because D8 is on the favorable doubling-down path (D8 → D4 → D2 → D1), while D14 offers no such path (missing D14 inside leaves 14, requiring S6, D4—a two-dart recovery). The difference is subtle but compounding: over hundreds of checkout attempts, the player who chooses routes ending on favorable doubles will finish more legs.

The Probability of a Nine-Dart Finish

The nine-dart finish—the perfect 501 leg—requires nine consecutive accurate throws: six treble 20s (scoring 180, 180) and then a checkout of 141 (commonly T20, T19, D12). What is the probability that a professional player achieves this?

A top professional hits treble 20 approximately 45–50% of the time when aiming at it. The probability of hitting six consecutive treble 20s is therefore roughly 0.47^6, which is about 1.1%. The checkout of 141 requires hitting T20 (~47%), then T19 (~45%), then D12 (~40%). The probability of that sequence is about 0.47 × 0.45 × 0.40 = 8.5%. Multiply the two: 1.1% × 8.5% ≅ 0.09%, or roughly 1 in 1,100 legs.

This aligns reasonably well with observed data. Top professionals play several thousand legs per year in competition, and the best among them hit a televised nine-darter every year or two. The calculation is approximate—it does not account for pressure, fatigue, or the specific dynamics of each throw—but it demonstrates that the nine-dart finish, while rare, is a predictable statistical event at the highest level.

For a club player hitting treble 20 about 15% of the time, the same calculation gives a probability of roughly 1 in 70 million legs. This is why amateur nine-darters are essentially unheard of.

How Professionals Use Math Intuitively

Professional darts players do not perform explicit probability calculations at the oche. But they have internalized the mathematical principles through tens of thousands of hours of practice and competition. They know, without computing, that leaving 32 (D16) is better than leaving 36 (D18). They know that leaving an odd number on 2 darts is bad because it eliminates the possibility of a direct double finish. They know that aiming at T19 for a 57 setup leaves 24 (D12), while aiming at T20 for 60 leaves 21 (which requires S5, D8—an extra dart).

This intuitive math is most visible in the endgame. Watch a professional player on a score of 87. Within a fraction of a second, they have decided on T17 (51) to leave 36 (D18), or perhaps S19 to leave 68, followed by T18, D7. The choice depends on their personal strengths—which trebles they hit most consistently, which doubles they favor. The best players have memorized checkout paths for every score from 2 to 170 and can adjust in real time when their first dart misses.

Statistics bear this out. At the top level, three-dart averages have risen from around 90 in the 1990s to above 100 today, with the very best players averaging 105–110 over entire tournaments. Checkout percentages have also climbed, with top players converting around 40–45% of their double attempts. These improvements reflect not just better throwing technique but a deeper collective understanding of optimal strategy—the mathematical game behind the physical game.

Practical Takeaways for Your Own Game

1. Know your accuracy. If you regularly miss the treble 20 and land in 1 or 5, you will score higher aiming at treble 19. Try both over 100 darts and compare your averages.
2. Memorize key checkouts. You do not need all 161. Focus on the scores you see most often: 32 (D16), 40 (D20), 36 (D18), 24 (D12), and the common two-dart finishes like 50 (S18, D16), 56 (T16, D4), and 80 (T20, D10).
3. Prefer D16 paths. When you have a choice of checkouts, choose the one that leaves D16, D8, or D4. The doubling-down path forgives inside misses.
4. Set up even numbers. Always try to leave an even number after your first dart in a checkout attempt. An odd remaining score means you cannot finish with a single double.
5. Do not chase 180s. Three treble 19s (171) with a tighter grouping is worth more over time than two treble 20s and a 5 (125). Consistency beats peaks.