Bayes rule and other problems

Bayes rule states that


Given one set of conditional probabilities, it enables one to calculate conditional probabilities with the reverse conditioning. However, we shall focus on the fact that Bayes rule enables one to calculate other probabilities when three probabilities (in this case: P(B|A), P(B|A'), and P(A); P(A')=1-P(A)) are given.

A Venn diagram for two events divides the sample space into four disjoint subsets: AB, A'B, AB', A'B'. The probabilities of these four events can be concisely represented with a square:

               A      A'                           
            |      |      |
         B  |  x   |  y   |  x+y=P(B)              
            |      |      |
         B' |  z   |  w   |  z+w=P(B')                              
              x+z=   y+w= |
              P(A)   P(A')|  1
In accordance with the row and column labels, this square means that P(AB)=x, P(A'B)=y, P(AB')=z, and P(A'B')=w. P(A)=x+z and P(B)=x+y as indicated above. P(A|B) = P(AB)/P(B) = x/(x+y); and the other conditional probabilities can be represented in a similar manner.

There are four unknowns (x, y, z, and w) in the above square, in terms of which all the probabilities we are interested in can be calculated. One constraint is that x+y+z+w=1 (P(S)=1); hence it is reasonable that three further equations would enable us to solve for x, y, z, and w, hence all probabilities.


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