calculation expected score of series of shots at a target with several scoring zones

andrei186

New member
Joined
Jan 19, 2018
Messages
4
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:

  1. If the above algorithm for 10 hits out of 10 shots is correct?
  2. If it is, how to handle the case with 1 miss?
 
The link above is not working (blank white page). Can you fix it?
Thank you for letting me know. Indeed, Firefox says the image contains an error. Uploaded it once again and checked in Firefox and Amigo - it shows.
BTW, I placed no link on the word 'function' in the same paragraph - it looks like it was added somehow automatically
 
… Uploaded it once again and checked in Firefox and Amigo - it shows.
Do you need to edit the link, too? It's still broken.


… I placed no link on the word 'function' in the same paragraph - it looks like it was added somehow automatically
That's a feature of this forum; certain math terms are auto-linked to corresponding lesson pages located on the main site (i.e., outside the forum).
 
Here is an illustration of what I would like to achieve:
http://xn----dtbqvlcm.xn--p1ai/images/87a.JPG

This is the 14cm x 14cm target with 10 scoring zones.
10 shots landed within zone 6 (according to the scoring rules, shot No4 - the one at the bottom of the target - is counted as 6 even if it just touches zone 6).
This means that all the 10 shots would hit a round silhouette target of 7 cm in diameter.

So I could expect that if I have 10 hits out of 10 shots at a round silhouette target of 7 cm in diameter, this with a good probability be equivalent of 87 points on the 14cm x 14cm target.

This is for 10 hits out of 10 shots.
I would like to calculate similar expected scoring points for 9 hits out of 10 shots, 8 hits out of 10 shots etc.
 
Top