Lager-swilling boffins crack secret of darts
Gaussian probability distribution points to better 'arrers'
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Anyone wishing to give Eric "The Crafty Cockney" Bristow a run for his money at the oche has been offered a mathematical helping hand by a crack team from Aberdeen University, the Daily Telegraph reports.
The group, from the Department of Plant and Soil Science, has demonstrated that while the optimum aiming point for skilled players is the treble-20, less fortunate "arrers" aficionados should try and hit the treble-19, with those even more cack-handed better off going for the bullseye.
The end result is that "if a player's aiming skill is not as good as a professional's, then aiming at the other parts of the dartboard would ensure the mean score will be higher over time".
First up, the team created "a computer digital dartboard, along with virtual players [and] carried out endless simulations of a range of skill levels to figure out the spread of dart impacts". They worked on the assumption that "the dart would land randomly within...a Gaussian probability distribution - or a bell-shaped curve".
For the aforementioned Crafty Cockney, the bell is "tall and thin, reflecting how the darts would cluster around a narrow spot". For mere mortals, the bells flattens as the player's skill decreases, indicating increasingly widespread impact points.
Naturally, the Aberdeen Uni researchers then needed to prove their theory with some solid time at the coalface, viz: their local boozer the St Machar Bar.
Dr Matthew Aitkenhead, who studies the effects of climate change on soil when not swilling lager in a nylon shirt, explained: "We pinned paper on a darts board and threw darts at it, measuring the distribution."
Aitkenhead confirmed: "As the player's performance is decreased there was a slight movement of the optimal target location upwards from the treble-20, followed by a large jump towards treble-19. From here, the optimal target location moves upwards and slightly left, before curving round towards the bull."
The findings make a good deal of sense, we reckon, given that missing the 20 by a narrow, but critical, margin offers the possibility of hitting a one or a five, while bunched around the 19 are three and seven. Less accurate players should then shift their aiming point left from the 19 where (going in a clockwise direction), the 7, 16, 8, 11 and 14 can be found (average 11.2). To the right, are 3, 17, 2, 15 and 10 (average 9.4). And if you're completely inept, aiming at the bull increases your chances of actually hitting the board at all.* ®
Bootnote
*That's our analysis, anyway. We're quite happy to stand corrected on this.
COMMENTS
Bivariate distribution
(1) Rayleigh: certainly not. Who said independent distributions horizontally and vertically? Bad idea. Easy to see your arm, body and the entire throwing movement is not split into two independent dimensions. All the little bits, joints, muscles and sinews conspire to mix up the two. (Carefully follow the path of a laser pointer in a trembling hand, for example.)
And what do you get if you mix up a lot of causes? The Gaussian, indeed mr. Kolmogorov.
(1b) Hitting the aim point zero probability: yes, duh, that holds for any point in any continuous distribution --- yet some point is hit in the end, yes? If all points have zero probability, nothing must be hit. Hm... flaw.
It's that pesky infinity thing. Ah, maybe we're not dealing with single dimensionless points in space?
(2) Any darts player knows the spread is easier to control horizontally than vertically, so squishing the bivariate `elliptic' curve into a univariate `round' Gaussian is a sin. The article sounds as if they've done just that, `measure distance' and not the vector of horizontal and vertical distance, but that's not proof this is what they did.
That said, the `Letters' submitters should have been pointed out their obvious mistakes --- one mixes `gaussian' with `uniform' distribution and misses the whole plot, the next could have been hinted that 2D gaussians give exactly the ellipses of equiprobability that he experiences.
Certainly since the `Letters' column was in the hands of the science editor!
Darts Experiment?
Re gausian Disribution
This is Aberdeen in winter what other excuse do you need to go to the pub.rotflmao.
Isotropic Gaussian?
It looks from your description of bell curves, etc. that the distribution they're using is essentially 1D - and that will translate into an isotropic (circular) 2D distribution. Surely, when aiming through a beery haze, the lefty-righty bit is easier than the uppy-downy bit. The Gaussian should be taller than wider, when seen projected onto the vertical dartboard. It might even have a bit of a slant to it. I find double 11s easier to hit than double 20s, for example. I think an anisotropic scatter would change their results somewhat, and they haven't even started to factor in Monte Carlo simulations of how likely a finish is in a game of 501. Also, where's the Plant and Soil Science angle?
..I'll get me coat.
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