Monte Carlo integration is a simple and general way to compute a (conditional) probability, expected value, etc.

Different ways to sample from the unit disk:

• Uniform x,y, then reject if too far from center.
• Uniform x,y, then scale and accept (or weight) by probability (or importance) max(|sin θ|, |cos θ|).
• Uniform r,θ, then accept (or weight) with probability (or importance) r.
• Uniform r²,θ, then compute r = √r².
• Uniform u,v,θ, then compute r = max(u,v).
• Stratify by dividing the range of say θ into intervals?

Different ways to sample sun rays:

• Uniform direction (from a sphere), then uniform offset (from a disk).
• Uniform surface point (from a sphere), then cosine direction (from a hemisphere, by picking, rejection, or importance).
• Work backwards from the eye!