Where should a darts player aim in order to maximise the score?

If we assume that the accuracy of a player who aims at (0, 0) is given by the classic bell curve:

P(x, y) δx δy

=

1 π λx λy

e (x/λx)2 e (y/λy)2 δx δy

             where P(x, y) δx δy is the probability of hitting an area δx δy located at (x,y).
            λ_x measures the player's accuracy (or lack of accuracy) in the horizontal direction,
            and λ_y measures the player's accuracy (or lack of accuracy) in the vertical direction,
            We assume that λ_y > λ_x.

∫

∞ – ∞

dx

∫

∞ – ∞

dy

P(x, y)

=

1

The probability of scoring double 1 when aiming at the centre of double 1 is:

∫

r2 r1

dr

∫

q 2 q 1

dθ

P(x, y)

             where x   =   r cos θ   –   ½ (r₁ + r₂) cos [ ½ (θ ₁ + θ ₂) ]
             and y   =   r sin θ.   –   ½ (r₁ + r₂) sin [ ½ (θ ₁ + θ ₂) ]

Let θ ₀   =   ½ (θ ₁ + θ ₂)     and     r₀   =   ½ (r₁ + r₂)
Let φ   =   ½ (θ ₁ – θ ₂)     and     μ   =   ½ (r₁ – r₂)

The probability of scoring double 1 can be rewritten:

∫

r2 r1

dr

∫

q 2 q 1

dθ

P(x, y)

=

∫

+ μ - μ

dμ

∫

+ φ - φ

dφ

P(x, y)

             where x   =   ( r₀ + μ ) cos ( θ ₀ + φ )   –   r₀ cos θ ₀
             and y   =   ( r₀ + μ ) sin ( θ ₀ + φ )   –   r₀ sin θ ₀

Of course we must consider the possibility of aiming somewhere else. In general, we can use (r₀, θ ₀) as the polar co-ordinates of the point of aim, and the segment of board as having   θ ₁ &le θ &le θ ₂   and   r₁ &le r &le r₂.
The probability is:

∫

r2 r1

dr

∫

q 2 q 1

dθ

P(r cos θ   – r0 cos θ 0 ,   r sin θ   – r0 sin θ 0)