Validation # 1
The Darts Assistant depends
heavily on being able to calculate the probabilities of hitting various parts of the
board. Specifically, it has functions that can calculate the probability that a
player with a specific DA Rating will hit a specific segment if he aims at a specific
point on the dartboard. The logic of the entire program depends on these functions
being accurate for all possible DA Ratings, segments and target points.
Let’s take this case (chosen at random) –
DA Rating : 
70 
Segment : 
Double ‘1’ 
Target : 
The middle of Double ‘18’ 
For this specific example the
Darts Assistant’s routines estimate the probability as 0.02384356. In other
words, if a player with a DA Rating of 70 were to aim at the middle of Double
‘18’ the Darts Assistant estimates that he would have a 2.384356% chance
of actually hitting Double ‘1’ instead.
You can use a spreadsheet
application such as Microsoft Excel to verify that this is in fact a very accurate
estimate. Here’s one way to do that :

First, make yourself familiar with the notation and
terminology used in the
paper
.

Let the origin of the x and ycoordinates be the
centre of the dartboard. (So the xaxis runs through the middle of the
‘11’ and ‘6’ sectors, and the yaxis runs through the
middle of the ‘20’ and ‘3’ sectors.)

Let x_{t} and y_{t} be the
coordinates of the target point. Satisfy yourself that when the target point is
the middle of Double ‘18’
x_{t} = 6.4375 × cos(54º ) = 3.78386756 inches
y_{t} = 6.4375 × sin(54º ) = 5.20804690 inches

Next, imagine the dartboard divided into concentric
“microrings” centred on the middle of the dartboard, each ring being
1/16 th of an inch wide (0.0625”). The outer edge of the Double ring is
6.625” from the centre of the dartboard. So there are 106 = 6.625
÷ 0.0625 of these concentric rings.

Now imagine each of these microrings divided into
720 equal “microsegments”, each subtending an angle of 0.5º at
the centre of the dartboard. The microsegments are arranged so that each
microring in each sector (e.g. in the ‘20’ sector) contains exactly
36 = 18 ÷ 0.5 complete microsegments. There are therefore 720
microsegments in each microring.

Altogether, there are 76,320 = 106 × 720
microsegments on the dartboard.

Notice that the microsegments in the outermost
(106th) microring are almost square  about 1/16th inch on each side – but
they get progressively narrower as they get closer to the centre of the
dartboard.

As an example to illustrate these definitions,
let’s take the microsegment that is in the bottom right corner of Double
‘20’. Please satisfy yourself that the following statements are
correct:

Its righthand edge is at an angle of 81º with the xaxis.

Its lefthand edge is at an angle of 81.5º with the xaxis.

Its lower edge is 6.25 inches from the centre.

Its upper edge is 6.3125 inches from the centre.

It is in the 101^{st} microring, counting
from the centre of the dartboard.

Counting anticlockwise from the xaxis, it is the
163^{rd} microsegment in that 101^{st} microring.

Its midpoint is 6.28125 inches from the centre and
is at an angle of 81.25º, measured anticlockwise from the xaxis.

The parameter s of the bivariate normal distribution in this example is
σ 
= 
3.1  0.029 × DA Rating 

= 
3.1 – 0.029 × 70 = 1.07 

The density function of the target distribution (see
page 872 of the paper) is
that is,

Now consider one of the microsegments and let x and
y be the coordinates of its midpoint. Also, let r be the distance of this midpoint
from the centre of the dartboard. So,
The outer and inner edges of this
microsegment are respectively r + 0.03125 and r – 0.03125 inches from the
centre. Satisfy yourself that the area of this microsegment is:

Since the microsegment is very small we can assume
that the probability density is uniform over its entire area. Therefore, the
probability of a dart landing in this microsegment is equal to
(prob density at the midpoint) x (area of the microsegment)

Now lay out an array (or range) of cells in a
spreadsheet such as Excel that has 106 columns and 720 rows. Let each column
correspond to a microring. The cells in each column correspond to the
microsegments in that microring. Write a formula in each cell that represents
the probability of hitting the corresponding microsegment. (Hint: Use a Data
Table to do this.)

Finally, you can estimate the probability of hitting
Double ‘1’ (or any specific area of the board) by summing the
probabilities of hitting the microsegments that lie in that area of the
board.

You can download an Excel file that performs these
calculations if you rightclick (optionclick on the Mac) on this link
DACheck1.xls
, and then choose “Save Link As …”.

Recall the case we were considering of a player with
a DA Rating of 70 aiming at the middle of Double ‘18’. We said that
the Darts Assistant had estimated that he would have a 2.384356% chance of
actually hitting Double ‘1’. You should now be able to confirm that
the Darts Assistant’s estimate must be pretty accurate – in this case,
at least  because
DACheck1.xls
gives an estimate of 2.384733%.