12. Assume that the centre is uniform on the rectangle [0, a] × [0, b], and that the acute angle θ between the needle and a line of the first grid is uniform on [0, 1 2π]. There is no intersection if and only if the centre lies within an inner rectangle of size

(a − ℓ cos θ) × (b − ℓ sin θ). Hence, the probability of an intersection is 2

πab Z π/2 0



ab − (a − ℓ cos θ)(b − ℓ sin θ)



dθ =

2ℓ

πab

(a + b − 1 2 ℓ).