Bayes’ Theorem is one of the most elegant theorems in probability. It can be viewed in many ways. With reference to the previous post, it is a way of expressing the likelihood that the pen lands on the triangle A given that it will land on rectangle B in terms of the likelihood that the pen lands on the rectangle B given that it will land on the triangle A, the probability that the pen will land on triangle A and the probability that the pen lands on the rectangle B.
The magic of this theorem is not felt unless one considers its application in spam filtering. Think of it: you can express the probability that an email is spam, given that it has vulgar words in it, in terms of the probability of finding those vulgar words in spam email, the probability that any email is spam and the probability of finding those words in any email.
The derivation is simply some airthemetic:
The actual philosophical derivation and a more verbal description of the counter-intuitive nature of this theorem (which by the way, guides the clock work behind the human mind) can be found here.