Our Solution
Problem: Dart players often do not chose the optimal targets toward the end of '01 games.
Reason: It is incredibly difficult to quantify a dart player's throw variances and translate them into a complete and probabilisticly correct analysis.
Solution: "Know Your Outs" solves 3 key hurdles to computing the correct out:
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Profile Measurement - We take throw results from specified targets around the board and compute each dartist's unique throw profile. Using a proprietary version of expectation maximization hypothesis techniques, we can take resulting region scores and fit them to bi-variate normal distributions with variances in terms of millimeters. Imagine a 3 dimensional bell over the target, the highest point of the bell with the largest area under the bell indicates the highest probability results. With larger and larger misses getting a smaller and smaller probability (i.e smaller area under the bell). Every dartist has a bell distribution such as this for any particular target; Some dartist's bell is squished width-wise, while others are flattened horizontally, and most have some degree of twisting to their bell. We measure these parameters exactly.
- Region Hit Probability - Using these statistics we can compute, for every target on the board, the area above each scoring region under the bell; which represents the probability of landing in that area of the board. Traditionally this is done with double integration from calculus. We use a mathematically equivalent proprietary technique that uses Fourier transforms and convolutions. It makes the computations much faster when using matrices, but it is basically just a very sophisticated way of computing the area under the bell for every possible target and each resulting region when aiming at those targets. This gives us a probability (however small some of the numbers may be) of landing in any region of the board for any target.
- Closing Probability - Using what we know about throw results from each target. We start with the most elementary out and work our way up. We start with a score of 2 and 1 dart in hand. Evaluate every target on the board and store the result with the highest probability of closing. We then run the analysis for a score of 2 and 2 darts in hand. If we don't win or bust with the first dart we use the results we have already found for a score of 2 and 1 dart in hand. We continue this recursive process using the answers for lower outs when addressing higher outs. We incrementally add darts in hand starting at one dart in hand and also add full turns to each of those scenarios. The lower outs understandably are not too computationally complex but the branches of permutations increase geometrically as we get to higher and higher outs. All the time, applying fundamental laws of probability, we consider every target around the board and compute the probability of closing. The heat maps are the summary result of this process and indicate the relative attractiveness of every target on the board.
This project took five years of research, coding, and testing. Mathematics included; exceptions maximization hypothesis, double integral calculus, bi-variate distributions, Fourier transforms, convolutions, Cartesian as well as polar spaces and their translations, and a very heavy use of probability. We've had to use multiple coding programs and matrix computation techniques to accommodate all the requisite tasks, and we built servers and and a computing cluster to manage the computationally complex workload. In short, we did not compromise and/or take any shortcuts in our endeavor to present the most complete and accurate analysis of outs to date. We are hopeful that you find the output and our service useful. Click the image below to download an example of our revolutionary heat maps. Also, please visit us on YouTube for more thoughtful discussion and examples.
Thank you for your support.
Know Your Outs Team
