How is Bayes’ Rule used to compute conditional probability?
We can solve for either conditional probability, but if we solve for P(B | A) we get the most basic form of Bayes’ Rule.
This expression allows us to compute one conditional probability in terms of the “reverse” conditional probability. In practice, the most challenging part of using Bayes’ Rule is identifying the events and computing the probabilities on the right side. We can simplify this task using a tree diagram.
This tree diagram is defined in terms of the marginal probabilities P(E) and P(E’), as well as the conditional probabilities P(R | E), P(R’ | E), P(R | E’), and P(R’ | E’). If we want to find the likelihood of one of these conditional probabilities reversed such as P(E | R), we apply Bayes’ Rule to give
Example 6 Bayes’ Rule
Using the events
E: return is selected for further examination
E’: return is not selected for further examination
R: return results in a refund of taxes paid
R’: return does not result in a refund of taxes paid
find the probability that a return was examined if we already know it resulted in a refund.
Solution We want to find P(E | R). Since the tree diagram is drawn to correspond to events given E or E’, we’ll apply Bayes’ Rule to “reverse” the conditional probabilities. For this conditional probability, Bayes’ Rule gives us
The probabilities in the numerator are located along the green branch. The probability in the denominator is found using the red and green branches which all terminate at R. Put the numbers in to yield
If a return results in a refund, it is unlikely the return was examined.