Imagine we drop a ball from some height onto the ground, where it only has one dimension of motion. How likely is it that a ball will go a distance c if you drop it and then drop it again from above the point at which it landed?
Letâ€™s break this down. After the first drop, it will land a units away from the starting point with probability f(a), where f is the probability distribution.
Now after this first drop, we pick the ball up and drop it from another height above the point where it first landed. The probability of the ball rolling b units away from the new starting point is g(b), where g may be a different probability distribution if itâ€™s dropped from a different height.
If we fix the result of the first drop so we know the ball went distance a, for the ball to go a total distance c, the distance traveled in the second drop is also fixed at b, where a+b=c. So the probability of this happening is simply f(a)â‹…g(b).1