Bayes' Theorem

Bayes' Theorem

However, just for the sake of argument, let's say that you want to know what Bayes' formula is.
Let's use the same example, but shorten each event to its one letter initial, ie: A, B, C, and D instead of Aberations, Brochmailians, Chompieliens, and Defective.
P(D|B) is not a Bayes problem. This is given in the problem. Bayes' formula finds the reverse conditional probability P(B|D).
It is based that the Given (D) is made of three parts, the part of D in A, the part of D in B, and the part of D in C.

                            P(B and D)
   P(B|D) =  -----------------------------------------
              P(A and D)  + P(B and D)  + P(C and D)

Inserting the multiplication rule for each of these joint probabilities gives

                            P(D|B)*P(B)
   P(B|D) =  -----------------------------------------
              P(D|A)*P(A) + P(D|B)*P(B) + P(D|C)*P(C)

However, and I hope you agree, it is much easier to take the joint probability divided by the marginal probability. The table does the adding for you and makes the problems doable without having to memorize the formulas.