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 :

  1. First, make yourself familiar with the notation and terminology used in the paper .
  2. Let the origin of the x- and y-coordinates be the centre of the dartboard. (So the x-axis runs through the middle of the ‘11’ and ‘6’ sectors, and the y-axis runs through the middle of the ‘20’ and ‘3’ sectors.)
  3. Let xt and yt be the coordinates of the target point. Satisfy yourself that when the target point is the middle of Double ‘18’
    xt = 6.4375 × cos(54º ) = 3.78386756 inches
    yt = 6.4375 × sin(54º ) = 5.20804690 inches
  4. Next, imagine the dartboard divided into concentric “micro-rings” 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.
  5. Now imagine each of these micro-rings divided into 720 equal “micro-segments”, each subtending an angle of 0.5º at the centre of the dartboard. The micro-segments are arranged so that each micro-ring in each sector (e.g. in the ‘20’ sector) contains exactly 36 = 18 ÷ 0.5 complete micro-segments. There are therefore 720 micro-segments in each micro-ring.
  6. Altogether, there are 76,320 = 106 × 720 micro-segments on the dartboard.
  7. Notice that the micro-segments in the outermost (106th) micro-ring 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.
  8. As an example to illustrate these definitions, let’s take the micro-segment that is in the bottom right corner of Double ‘20’. Please satisfy yourself that the following statements are correct:
  9. 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
  10. The density function of the target distribution (see page 872 of the paper) is
    Formula 1
    that is,
    Formula 2
  11. Now consider one of the micro-segments 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,
    Formula 3
    The outer and inner edges of this micro-segment are respectively r + 0.03125 and r – 0.03125 inches from the centre. Satisfy yourself that the area of this micro-segment is:
    Formula 4
  12. Since the micro-segment 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 micro-segment is equal to
    (prob density at the midpoint) x (area of the micro-segment)
  13. 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 micro-ring. The cells in each column correspond to the micro-segments in that micro-ring. Write a formula in each cell that represents the probability of hitting the corresponding micro-segment. (Hint: Use a Data Table to do this.)
  14. Finally, you can estimate the probability of hitting Double ‘1’ (or any specific area of the board) by summing the probabilities of hitting the micro-segments that lie in that area of the board.
  15. You can download an Excel file that performs these calculations if you right-click (option-click on the Mac) on this link DACheck1.xls , and then choose “Save Link As …”.
  16. 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%.