The randomized approach is based on the recognition that the formula defining the form factors can be taken to represent the probability of a quite simple event if the underlying probability distributions are defined properly.
An appropriate such event would be a surface j being hit by a particle leaving surface i. Let us denote the event that a particle leaves surface by . Expressing the probability of the ``hit'' of surface j by the total probability theorem we get:
The hitting of surface j can be broken down into the separate events of hitting the various differential elements composing . Since hitting of and hitting of are exclusive events if :
Now the probability distributions involved in the equations are defined:
where is the indicator function of the event `` is visible from ''.
Substituting these into the original probability formula:
This is exactly the same as the formula for form factor . This probability, however, can be estimated by random simulation. Let us generate n particles randomly using uniform distribution on the surface i to select the origin, and a cosine density function to determine the direction. The origin and the direction define a ray which may intersect other surfaces. That surface will be hit whose intersection point is the closest to the surface from which the particle comes. If shooting n rays randomly surface j has been hit times, then the probability or the form factor can be estimated by the relative frequency:
Two problems have been left unsolved:
Addressing the problem of the determination of the necessary number of attempts, we can use the laws of large numbers.
The inequality of Bernstein and Chebyshev [Rén81] states that if the absolute value of the difference of the event frequency and the probability is expected not to exceed with probability , then the minimum number of attempts (n) is:
The generation of random distributions can rely on random numbers of uniform distribution in [0..1] produced by the pseudo-random algorithm of programming language libraries. Let the probability distribution function of the desired distribution be P(x). A random variable x which has P(x) probability distribution can be generated by transforming the random variable r that is uniformly distributed in [0..1] applying the following transformation: