prhw6ob5.htm - s rule and other problems Bayes rule states...

s rule and other problems
Bayes rule states that
P(A|B)=P(B|A)P(A)/(P(B|A)P(A)+P(B|A')P(A'))
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