When shooting at a double target of two croquet balls with a gap that your ball could go through in the middle, is it better to aim for the middle or for one of the balls? The answer depends on the size of the gap, and on the probability distribution of the directional error in your shot.
If the gap is narrow enough, you are best aiming at the middle (assuming that your error distribution is symmetrical about 0), since this gives you the maximum probability of not being wide of the target, and the chance of going through the gap is very small. How narrow is narrow enough depends on the dispersion and shape of your error distribution: if you have a high probability of being almost exactly on line then the risk of going through the gap is increased and you may be better aiming for one of the balls. More precisely, if your distribution is a mixture of one or more zero-mean normal distributions or something approximating this, you may be better aiming at a point slightly off-centre on one of the balls, towards the side where the other ball is, since this increases your (still relatively small) chance of hitting the second ball, with only a small reduction in your probability of hitting the first one. The more leptokurtic the error distribution is (at a given variance), the sooner aiming at the middle of the double target ceases to be optimal as the gap size is increased. [Actually that last sentence is not quite accurate, since there will be exotic distribution shapes for which the critical gap size is not strictly decreasing as the kurtosis increases, but it should hold true for a realistic class of "well-behaved" distribution shapes.]
So far I have discussed the problem in relation to what is effectively a one-dimensional target (assuming that the striker's ball remains on the ground). But similar principles apply to a two-dimensional target, where there are horizontal and vertical components in the error. I think the appropriate generalisation is to targets that are non-convex (i.e. in which all points on the straight line between a given pair of points within the target are not necessarily themselves within the target). In the one-dimensional case, being non-convex is equivalent to having at least one gap. For a convex target (which in the one-dimensional case means a target without a gap), aiming at the middle is always optimal, at least in one dimension; I think, but have not proved, that in more than one dimension the optimal aiming point is the centroid of the target. [PS [2012-03-10] On further thought I realise that it's not always the centroid. Consider a very tall narrow isosceles triangle: the centroid is one third of the way from base to apex, but I'm pretty sure the optimal aiming point will be less far up, near the incentre, at least when the RMS aiming error is much smaller than the height of the triangle.]
In all this, I have implicitly taken account of the non-zero width of the projectile (the striker's ball). Since hitting the target with any part of the striker's ball counts as success, the problem is actually equivalent to aiming with a zero-width projectile at a target whose every component is extended by half a ball width on each side. Accordingly it is only if the space between two target balls is more than a ball's width that it counts as a gap: if it's narrower, the striker's ball can't get through it.
PS [2011-08-06] David Appleton and Campbell Morrison have now implemented a mathematical model of single-target and double-target shots and an interactive probability calculator based on it: see www.scottishcroquet.org.uk/roquet/. Looking at double targets with different separations, and plotting roquet probability against aiming point, confirms that aiming off-centre on one of the balls is optimal when the gap is wide enough, though it's never much better than aiming at the centre of the ball.
Attention to detail
[Page created: 2009-07-16. Page modified: 2012-03-13.]
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