How Likely Are You to Hit the Centre of the Archery Target?
Hitting the archery target:
Gunnar Richter - Wikimedia Commons
Archery
Archery was historically used primarily for hunting and warfare, but is now a popular sport practised around the world. It has been an Olympic sport since 1900, appearing at the Olympic Games in 1900, 1904, 1908, 1920 and then at every Olympics from 1972 to the present day. At the next Olympics in Paris 2024, there will be five archery events covering men's and women's individual and team events, and the mixed team event. It is the Olympic Archery rules that we will be using in the article.
An archery target:
Alberto Barbati - Wikimedia Commons
The rules of Olympic archery
Olympic archers use a recurve bow (a bow where the strings are attached directly to the ends of the bow and the archer aims through a sight in the bow) and fire at a target placed 70 metres away.
The target is a circle with a diameter of 122 cm, split into 10 evenly spaced concentric rings (plus an extra circle inside the inner circle which we will discuss later). As the radius of the target is 122 ÷ 2 = 61 cm and there are ten rings, then working outwards from the centre of the target, each ring is 6.1 cm thick.
Different points are scored depending upon which ring is hit by the arrow. The outermost white ring scores 1 point, the inner white ring score 2 points, the outer black ring scores 3 points and so on until we reach the inner gold circle which scores the full 10. Each archer shoots a set number of arrows and the scores are added up. Whoever scores the most is the winner.
The smaller circle inside the centre 10 circle (sometimes known as the X ring) is used in the case of a tie-break. If two archers have the same number of points, the archer with most arrows in the X ring is the winner.
The target is a circle with a diameter of 122 cm, split into 10 evenly spaced concentric rings (plus an extra circle inside the inner circle which we will discuss later). As the radius of the target is 122 ÷ 2 = 61 cm and there are ten rings, then working outwards from the centre of the target, each ring is 6.1 cm thick.
Different points are scored depending upon which ring is hit by the arrow. The outermost white ring scores 1 point, the inner white ring score 2 points, the outer black ring scores 3 points and so on until we reach the inner gold circle which scores the full 10. Each archer shoots a set number of arrows and the scores are added up. Whoever scores the most is the winner.
The smaller circle inside the centre 10 circle (sometimes known as the X ring) is used in the case of a tie-break. If two archers have the same number of points, the archer with most arrows in the X ring is the winner.
The areas of each ring
Before we can work out any probabilities, we need to look at the areas of each ring on the target.
We mentioned before that working from the centre of the target, each ring is 6.1 cm wide. Using this information, it is simple enough to find the area of the inner 10 circle. The area of a circle can be found using the formula A = π r^2 where r is the radius. The area of the inner 10 circle is therefore π × 6.1^2 = 37.21 π = 116.90 cm^2 (to 2 decimal places).
To find the area of the outer gold ring, we think of this as a circle of radius 12.2 cm with the inner gold circle cut out from the middle. Therefore:
area of outer gold ring = π × 12.2^2 − π × 6.1^2 = 111.63 π = 350.70 cm^2.
Similarly the inner red ring is a circle of radius 3 × 6.1 = 18.3 cm minus the full gold circle of radius 12.2 cm.
area of inner red ring = π × 18.3^2 − π × 12.2^2 = 186.05 π = 584.49 cm^2.
By following the same rules we can find the areas of each ring as in the following table.
We mentioned before that working from the centre of the target, each ring is 6.1 cm wide. Using this information, it is simple enough to find the area of the inner 10 circle. The area of a circle can be found using the formula A = π r^2 where r is the radius. The area of the inner 10 circle is therefore π × 6.1^2 = 37.21 π = 116.90 cm^2 (to 2 decimal places).
To find the area of the outer gold ring, we think of this as a circle of radius 12.2 cm with the inner gold circle cut out from the middle. Therefore:
area of outer gold ring = π × 12.2^2 − π × 6.1^2 = 111.63 π = 350.70 cm^2.
Similarly the inner red ring is a circle of radius 3 × 6.1 = 18.3 cm minus the full gold circle of radius 12.2 cm.
area of inner red ring = π × 18.3^2 − π × 12.2^2 = 186.05 π = 584.49 cm^2.
By following the same rules we can find the areas of each ring as in the following table.
Ring |
Calculation |
Area in terms of pi (cm^2) |
Area as a decimal (cm^2) |
Inner Gold 10 Points |
π × 6.1^2 |
37.21 π |
116.90 |
Outer Gold 9 Points |
π × 12.2^2 − π × 6.1^2 |
111.63 π |
350.70 |
Inner Red 8 Points |
π × 18.3^2 − π × 12.2^2 |
186.05 π |
584.49 |
Outer Red 7 Points |
π × 24.4^2 − π × 18.3^2 |
260.47 π |
818.29 |
Inner Blue 6 Points |
π × 30.5^2 − π × 24.4^2 |
334.89 π |
1052.09 |
Outer Blue 5 Points |
π × 36.6^2 − π × 30.5^2 |
409.31 π |
1285.89 |
Inner Black 4 Points |
π × 42.7^2 − π × 36.6^2 |
483.73 π |
1519.68 |
Outer Black 3 Points |
π × 48.8^2 − π × 42.7^2 |
558.15 π |
1753.48 |
Inner White 2 Points |
π × 54.9^2 − π × 48.8^2 |
632.57 π |
1987.28 |
Outer White 1 Point |
π × 61^2 − π × 54.9^2 |
706.99 π |
2221.07 |
Whole Target |
π × 61^2 |
3721 π |
11 689.87 |
The probabilities of hitting each ring
Now we know the areas of each ring on the target, we can calculate the probabilities of hitting each ring. It is important to note that these calculations are only valid for randomly fired arrows; they do not take into account the skill of the archer. Obviously in reality, an Olympic archer has a significantly higher chance of hitting the centre than a novice archer does. With these calculations we are going to assume the arrows are fired randomly, but are still guaranteed to hit the target somewhere.
Calculating the probabilities
If we assume a randomly fired arrow is going to hit the target, and all areas of the target are equally likely to be hit, then the probability of the arrow hitting a particular ring is:
area of the ring ÷ total area of target × 100
So the probability that the arrow hits the centre circle is 37.21π ÷ 3721π × 100 = 1%.
Note that for future calculations, as both the ring area and total area are multiples of π and we are dividing them, π will cancel, hence we can disregard it from the calculation.
The probability that the arrow hits the outer gold ring = 111.63 ÷ 3721 × 100 = 3%.
All of the probabilities can be calculated as such and are shown in the table below.
area of the ring ÷ total area of target × 100
So the probability that the arrow hits the centre circle is 37.21π ÷ 3721π × 100 = 1%.
Note that for future calculations, as both the ring area and total area are multiples of π and we are dividing them, π will cancel, hence we can disregard it from the calculation.
The probability that the arrow hits the outer gold ring = 111.63 ÷ 3721 × 100 = 3%.
All of the probabilities can be calculated as such and are shown in the table below.
Ring (points score) |
Probability of being hit by random arrow (%) |
10 |
1 |
9 |
3 |
8 |
5 |
7 |
7 |
6 |
9 |
5 |
11 |
4 |
13 |
3 |
15 |
2 |
17 |
1 |
19 |
So we can see from our calculations, that a randomly fired arrow that hits the target has a 1% chance of hitting the centre gold circle and scoring 10 points, all the way up to a 19% chance of hitting the outer white ring for 1 point. But what does this mean in practice?
An arrow that randomly hits the target has a 1% chance of hitting the centre circle and scoring 10 points.
What is the predicted score after 70 arrows?
We can calculate a predicted score using our probabilities. The qualifying score for the Olympics is 640 for men and 605 for women after 70 arrows. Let's calculate our predicted score after 70 arrows.
We will look at one arrow to start with. The predicted score can be found by multiplying each score by its probability (in decimal form) and adding all ten results together.
Predicted score for one arrow = 10 × 0.01 + 9 × 0.03 + 8 × 0.05 + 7 × 0.07 + 6 × 0.09 + 5 × 0.11 + 4 × 0.13 + 3 × 0.15 + 2 × 0.17 + 1 × 0.19 = 3.85
As we fire arrows randomly at the target, assuming they all hit the target somewhere, we would expect our average score per arrow to be 3.85.
Using this expected score per arrow, our expected score after 70 arrows would simply be 70 × 3.85 = 269.5 (269 or 270 as we can't score half-points). Taking into account that this calculation assumes all our arrows hit the target somewhere (which is definitely not guaranteed), hence we would probably score even lower, then we can see that we certainly have a lot of practice to do before we can achieve Olympic standard scores.
We will look at one arrow to start with. The predicted score can be found by multiplying each score by its probability (in decimal form) and adding all ten results together.
Predicted score for one arrow = 10 × 0.01 + 9 × 0.03 + 8 × 0.05 + 7 × 0.07 + 6 × 0.09 + 5 × 0.11 + 4 × 0.13 + 3 × 0.15 + 2 × 0.17 + 1 × 0.19 = 3.85
As we fire arrows randomly at the target, assuming they all hit the target somewhere, we would expect our average score per arrow to be 3.85.
Using this expected score per arrow, our expected score after 70 arrows would simply be 70 × 3.85 = 269.5 (269 or 270 as we can't score half-points). Taking into account that this calculation assumes all our arrows hit the target somewhere (which is definitely not guaranteed), hence we would probably score even lower, then we can see that we certainly have a lot of practice to do before we can achieve Olympic standard scores.
Comments
What's your best archery result? Is your score higher than the average? Don't forget to leave your comments below.