Conditional probability  (Page 5/6)

Reversal of conditioning

Students in a freshman mathematics class come from three different high schools. Their mathematical preparation varies. In order to group them appropriately in classsections, they are given a diagnostic test. Let H _i be the event that a student tested is from high school i , $1\le i\le 3$ . Let F be the event the student fails the test. Suppose data indicate

A student passes the exam. Determine for each i the conditional probability $P\left(H_i\right|F^c)$ that the student is from high school i .

SOLUTION

Then

Similarly,

A compound selection and reversal

Begin with items in two lots:

1. Three items, one defective.
2. Four items, one defective.

One item is selected from lot 1 (on an equally likely basis); this item is added to lot 2; a selection is then made from lot 2 (also on an equally likely basis). This second itemis good. What is the probability the item selected from lot 1 was good?

SOLUTION

Let G ₁ be the event the first item (from lot 1) was good, and G ₂ be the event the second item (from the augmented lot 2) is good. We want to determine $P\left(G_1\right|G_2)$ . Now the data are interpreted as

By the law of total probability (CP2) ,

By Bayes' rule (CP3) ,

Additional problems requiring reversals

• Medical tests . Suppose D is the event a patient has a certain disease and T is the event a test for the disease is positive. Data are usually of the form: prior probability $P\left(D\right)$ (or prior odds $P\left(D\right)/P\left(D^c\right)$ ), probability $P\left(T\right|D^c)$ of a false positive , and probability $P\left(T^c\right|D)$ of a false negative. The desired probabilities are $P\left(D\right|T)$ and $P\left(D^c\right|T^c)$ .
• Safety alarm . If D is the event a dangerous condition exists (say a steam pressure is too high) and T is the event the safety alarm operates, then data are usually of the form $P\left(D\right)$ , $P\left(T\right|D^c)$ , and $P\left(T^c\right|D)$ , or equivalently (e.g., $P\left(T^c\right|D^c)$ and $P\left(T\right|D)$ ). Again, the desired probabilities are that the safety alarms signals correctly, $P\left(D\right|T)$ and $P\left(D^c\right|T^c)$ .
• Job success . If H is the event of success on a job, and E is the event that an individual interviewed has certain desirable characteristics, thedata are usually prior $P\left(H\right)$ and reliability of the characteristics as predictors in the form $P\left(E\right|H)$ and $P\left(E\right|H^c)$ . The desired probability is $P\left(H\right|E)$ .
• Presence of oil . If H is the event of the presence of oil at a proposed well site, and E is the event of certain geological structure (salt dome or fault), the data are usually $P\left(H\right)$ (or the odds), $P\left(E\right|H)$ , and $P\left(E\right|H^c)$ . The desired probability is $P\left(H\right|E)$ .
• Market condition . Before launching a new product on the national market, a firm usually examines the condition of a test market as anindicator of the national market. If H is the event the national market is favorable and E is the event the test market is favorable, data are a prior estimate $P\left(H\right)$ of the likelihood the national market is sound, and data $P\left(E\right|H)$ and $P\left(E\right|H^c)$ indicating the reliability of the test market. What is desired is $P\left(H\right|E),$ the likelihood the national market is favorable, given the test market is favorable.