A particle is equally likely to lie anywhere on the circumference of a circle of radius R. Consider as the z axis any straight line in the plane of the circle and passing through its center. Denote by θ the angle between the z axis and the straight line connecting the center of the circle to a particle.

(a) What is the probability that the angle lies between θ and θ + dθ? Express your answer in terms of R, θ, and dθ.

(b) What is the probability that the particle’s z coordinate lies between z and z +dz? Express your answer in terms of R, z, and dz