Validation # 2
Let’s consider the case where a player with a DA Rating of 90 has a score of 2
at the start of his turn. Therefore he needs a double ‘1’ to go out. What
points should he aim at with his first, second and third dart?
This case is simple enough for it to be answered using a spreadsheet program.

Please refer to Fig. 2 in the
paper.
This figure shows the locations of the target points in a sector, and it assigns an index to
each target point. In our simple example, the only points a player might reasonably aim
at are the points 21, 22, 23 and 24 in the ‘1’ sector. To aim at any other
point would clearly be illogical.

For this example, the
Darts Assistant recommends that the player should aim at point 22 with his first two
darts, and point 21 with his third dart. (You can confirm this by playing an artificial
darts game on the app itself: set your DA Rating to 90 (by Direct Setting) then arrange
for your score to drop to 2.)

Use
DACheck1.xls
to estimate the
probability of hitting Double ‘1’ if the player aims at each of the four
possible target points 21, 22, 23 and 24. In the same way, estimate the probabilities
that he will miss the board (i.e. not score anything) if he aims at those four points.
This gives these results:
Target point # in the '1' sector

Distance from centre, r

Probability of missing the board

Probability of hitting Double '1'

21  6.4375  0.36492000  0.28669178 
22  6.625  0.51477347  0.26451904 
23  7.0  0.78868162  0.14818055 
24  7.375  0.94141227  0.04746863 

Now prepare a list of
all combinations of the four possible target points. For example, the player could aim
at point 21 on each of his three throws. Or he could aim at point 21 with his first
dart, point 21 with the second dart, and point 22 with the third dart. The list would
look like this:
1st target point

2nd target point

3rd target point

21  21  21 
21  21  22 
21  21  23 
21  21  24 
21  22  21 
21  22  22 
21  22  23 
21  22  24 
21  23  21 
21  23  22 
21  23  23 
21  23  24 
21  24  21 
21  24  22 
21  24  23 
21  24  24 
22  21  21 
22  21  22 
22  21  23 
22  21  24 

Etc.



Figure out for each
combination of target points the probability that the player will go out in one
turn.

You can download an
Excel file that performs these calculations if you rightclick (optionclick on
the Mac) on this link
DACheck2.xls,
and then choose “Save As …”.

Notice that this
workbook has two tabs: Unsorted and Sorted. They are identical except for the fact that
the Sorted sheet is sorted according to the probability of going out in one turn. You
can see that the best combination of target points is 22, 22 and 21 on the
1^{st}, 2^{nd} and 3^{rd} throws, respectively – which is
exactly what the Darts Assistant recommended.

Notice that even a
small change in the target points can make a big difference. If you aim at 22, 22, 21
you will go out in 2.10 turns, on average. If you aim at 21, 21, 21 (only
3/16^{th} of an inch difference on the first two darts) it will take you 2.33
turns – about 10% longer.