Shooting competitions are using a 14cm x 14cm target consisting of 10 concentric circles making ten scoring zones of 1 to 10 points:
The shots distribution across this target follows Gauss (normal) function and there are tables showing probability of hitting every of its 10 scoring zones, and corresponding graphs (Graph 1).
Graph 2 is built on Graph 1 and shows probability of scoring a particular zone or any other zone inside it.
I have a round silhouette target (without scoring zones) of 7 cm in diameter, which corresponds to zone 6 on the 14x14 target.
Now, shooting at my 7 cm target I would like using Graph 1 to calculate expected score of N hits out of 10 shots where 5<=N<=10, if these shots were placed on 14x14cm target.
For N=10 this is pretty steightforward.
10 hits means that all hits are within 7mm i.e. all the shots are either zone 6 or zone 7 ...or 10.
I.e. shots are distributed within thses five zones and their distribution follows Gauss law.
Therefore zones 6 to 10 should be superimposed on Graph 1.
This means that on Graph 1:
zone 10 should be merged with zone 9 giving probability of scoring 10: 6.0 + 16.1= 22.1%
zone 8 should be merged with zone 7 giving probability of scoring 9: 19.8 + 21.2= 41.0%
zone 6 should be merged with zone 5 giving probability of scoring 8: 16.8 + 9.5= 26.3%
zone 4 should be merged with zone 3 giving probability of scoring 7: 5.9 + 2.9= 8.8%
zone 2 should be merged with zone 1 giving probability of scoring 6: 1.2 + 0.6= 1.8%
Therefore expected score of 10 hits out of 10 shots at 7cm circle, if projected on 14 cm target, will be no less than 87.2 points:
10 points * 10 shots * 22.1% / 100 = 22.1 points
9 points * 10 shots * 41.0% / 100 = 36.9 points
8 points * 10 shots * 26.3% / 100 = 21.0 points
7 points * 10 shots * `8.8% / 100 = 6.1 points
6 points * 10 shots * 1.8% / 100 = 1.1 points
Total 87.2 points
However with one miss out of ten shots I stumbled.
Quesions:
The shots distribution across this target follows Gauss (normal) function and there are tables showing probability of hitting every of its 10 scoring zones, and corresponding graphs (Graph 1).
Graph 2 is built on Graph 1 and shows probability of scoring a particular zone or any other zone inside it.
I have a round silhouette target (without scoring zones) of 7 cm in diameter, which corresponds to zone 6 on the 14x14 target.
Now, shooting at my 7 cm target I would like using Graph 1 to calculate expected score of N hits out of 10 shots where 5<=N<=10, if these shots were placed on 14x14cm target.
For N=10 this is pretty steightforward.
10 hits means that all hits are within 7mm i.e. all the shots are either zone 6 or zone 7 ...or 10.
I.e. shots are distributed within thses five zones and their distribution follows Gauss law.
Therefore zones 6 to 10 should be superimposed on Graph 1.
This means that on Graph 1:
zone 10 should be merged with zone 9 giving probability of scoring 10: 6.0 + 16.1= 22.1%
zone 8 should be merged with zone 7 giving probability of scoring 9: 19.8 + 21.2= 41.0%
zone 6 should be merged with zone 5 giving probability of scoring 8: 16.8 + 9.5= 26.3%
zone 4 should be merged with zone 3 giving probability of scoring 7: 5.9 + 2.9= 8.8%
zone 2 should be merged with zone 1 giving probability of scoring 6: 1.2 + 0.6= 1.8%
Therefore expected score of 10 hits out of 10 shots at 7cm circle, if projected on 14 cm target, will be no less than 87.2 points:
10 points * 10 shots * 22.1% / 100 = 22.1 points
9 points * 10 shots * 41.0% / 100 = 36.9 points
8 points * 10 shots * 26.3% / 100 = 21.0 points
7 points * 10 shots * `8.8% / 100 = 6.1 points
6 points * 10 shots * 1.8% / 100 = 1.1 points
Total 87.2 points
However with one miss out of ten shots I stumbled.
Quesions:
- If the above algorithm for 10 hits out of 10 shots is correct?
- If it is, how to handle the case with 1 miss?