Some Remarks on Darts

Transcription

1 Some Remarks on Darts How simple mathematics can improve your score Vijay Tymms March 2013

2 Background and pre-empting some reactions This was some work I did when a PhD student at the University of Liverpool Possible question which might be at the back of you minds: How did he manage to get a PhD whilst doing stuff like this? Possible slightly incredulous reaction: Jeez, what a geek! It was a major chunk of procrastination that really delayed the write up Guilty as charged!

3 Aims of this work To show, using probability arguments, the best way to get a good score in a game of darts Demonstrate that casual players can improve their dart game Show that most amateur players have their tactics wrong Provide some simple instructions on how to improve to players at all levels

4 The dartboard

5 The standard game of darts Attain a score (usually 501) in as few darts as possible Must finish with a double or the bullseye Throw in turns with three darts per turn (max score 180) Two stages (i) amass enough points to get an outshot on a double (ii) finish the game by hitting a double This work concentrates on stage (i) how best to accumulate a high score to get within range of an outshot

6 Other variants Round the clock Killer Cricket These are all popular pub games One reason is that darts has a degree of luck to it!

7 Why darts is a popular pub game One reason is that because of the points distribution there is an unpredictable nature to a player s score On a single turn a poor player can score higher than a better player This makes the game popular and addictive. Like gambling, a poor player can win now and again against a better player Concentric circle darts wouldn t be as popular!

8 Why darts is a popular pub game

9 Suitability as a pub game A slightly silly graph Luck/Skill ratio, or, likelihood one player can beat a better player

10 An issue with casual players tactics Players often aim for the treble 20 to maximise their score The 20 is flanked by the 5 and the 1; the player risks getting the classic bed and breakfast score of 26 or worse

11 How the tactics may be improved An elite player is best off going for the treble 20 An absolute beginner is best off merely trying to hit the board they should aim right for the centre to try and get a score at all An intermediate player may be best off aiming somewhere else

12 Possible alternative locations

13 An idea to investigate the problem Write a computer program to investigate what average score a player will get if they aim for any part of the board It should cater for different levels of skill It should predict excellent players should aim for the treble 20 and poor players for the centre It should make testable predictions about where intermediate players should aim

14 An assumption: normal throwing distribution The model assumes that a player s shot follows a normal distribution in 2D The better the player the narrower the distribution The width of the curve can be calculated from the percentage of darts that land in the bullseye when aimed for

15 Some evidence for this The Badger s Arse Effect

16 Creating a pixellated dart board Splits a square into 200 x 200 pixels with a standard dart board in the centre and creates a contour plot:

17 Scanning the normal distribution over every pixel A player enters what percentage of darts land in the bullseye when aiming for it The program calculates the width of the normal distribution curve Program calculates what score is accrued by aiming at each pixel on the board

18 Some results

19 Some results

20 Some results

21 Some results

22 Some results

23 Some results

24 Some results

25 Some results

26 Assessing your own level 1: measure % in bullseye Two ways to see where it s best to aim 1. Throw lots of darts at the bull and see what percentage land in it The more darts you throw the more meaningful the percentage will be Let s try it anyway with just ten darts volunteer?

27 Test chart

28 Test chart

29 AND LET S SEE WHAT YOU COULD HAVE WON

30 It s a speedboat!

31 Assessing your own level 2: taking an average when aiming at the bull(*) 2. Aiming at the centre of the board, throw 10 darts and record your average score Refer to the test chart which matches you bullseye average? Volunteer? (*) Or anywhere actually, the same principle applies

32 Test chart

33 Test chart

34 Conclusions It is worth going for the treble 20 if you can get 3/10 shots or more in the bull If you get 1 or 2 out of 10 in the bull it is best to aim just below the treble 20 If 6-10% land in the bull there is a sweet spot around the treble 7 and 16 area that gets the highest score Less than this in the bull and you re better off aiming for the centre Most casual players probably fall into the latter category!

35 Some caveats The throw distribution may differ from the assumed one If you want a chance at beating a significantly better player than yourself the best option may be to aim for the highest score Aiming for the treble 20 may be a more enjoyable way of playing the game the pleasure of hitting it once in a while may override getting merely decent average scores Darts is a pub game and involves drinking. Does the best place to aim for change with increasing booze comsumption? You get worse with lots of drink but do a couple of beers actually help?!

36 Average score off 50 darts A thoroughly unhealthy experiment Cans of beer downed

37 A thoroughly unhealthy experiment

38 Possible further work Look at the possibility of skewed distributions or cater for an individual player Include bounce outs?! Consider out shots on doubles?

39 Acknowledgements I would like to thank Jim Bowen. Super smashing great