Math 166 Lecture Notes 1.7
Section 1.7 More Conditional Probabilites (A Posteriori Probabilities)
We have previously discussed the conditional probability of the occurrence of an event B,
given the occurrence of an earlier event A (a priori probabilities). Now we are going to
reverse the problem, and try to find the probability of an earlier event A conditioned on
the occurrence of a later event B (a posteriori probabilities).
We must recognize that there is a difference in the conditional probabilities P(A|B) and
P(B|A). As for as the probability computations, the latter is represented by a single
section of a branch on the tree diagram, while the other involves a more complicated
computation. In many textbooks, a rather foreboding formula, called Bayes’ Theorem, is
presented to compute the a posteriori probabilities.
However, below is a suggested approach for ALL conditional probabilities:
In either case, we could find the conditional probability by using the formulas:
)(
)(
)|( BP
BAP
BAP
∩
=, and, )(
)(
)|( AP
BAP
ABP
∩
=.
Note: For most examples in this section, give answers in decimal form, rounding to 4 places if necessary.
Example 1: Using the tree diagram below and the formulas shown above, compute
P(B|A) and P(A|B).
a) P(B|A) =
b) P(A|B) =
Example 2: Shops A, B, and C supply 50%, 30%, and 20%, respectively, of the cabinets
used by Carl’s Construction. Carl’s warehouse inventory manager has determined that
40% of the cabinets that were produced by Shop A are oak, and 70% of the cabinets
produced by Shops B and C are oak. Observe the tree diagram shown below.
a) Given that a randomly selected cabinet in
inventory is from Company B, what is the
probability that it is oak?
c) An oak cabinet is randomly selected. What is the
probability that it came from Shop B?
