Conditional Probability

Why do we need conditional probability? Gaining partial information relevant to the experiment’s outcome may cause us to revise the probability of the events.

Example Components are assembled in a plant that uses two assembly lines and . Line uses older equipment that is slower and less reliable. Suppose that on a given day has assembled 8 components, of which 2 are defective ( ) and six are nondefective ( ). Line has assembled 10 components, of which 1 is defective. One of the 18 components is chosen at random. The probability that the item came from line changes if we have the prior information that the selected item is defective.

New sample space In effect, knowing that event B occurs restricts the sample space to those outcomes that are in B. The outcomes that are of interest to event A are those in both A and B. Since the probabilities for simple events in B sum to P(B), we need to re-normalize probabilities by dividing by P(B).

Definition For two events A and B with P(B)>0, the conditional probability of A given that B has occurred is

Example Suppose that of all individuals buying a certain digital camera, 60% include an optional memory card, 40% include an extra battery, and 30% include both. Consider randomly selecting a buyer and let A=(memory card purchased) and B=(battery purchased). Let P(A)=0.6, P(B)=0.4 and P(both purchased)=0.3. We compute conditional probabilities for A given B and for B given A. Notice that they are different from the unconditional probabilities.

Multiplication rule

Law of total probability Let be mutually exclusive events whose union is the sample space (exhaustive). Then for any other event B,

Example An individual has three different email accounts. Of her messages, 70% come into account 1, 20% into account 2, and 10% into account 3. Of the messages into account 1, only 1% are spam, whereas the corresponding percentages are 2% and 5% for accounts 2 and 3. What is the probability that a randomly selected message is spam?

Bayes’ Rule Let be mutually exclusive and exhaustive events with prior probabilities, , . Then for any other event for which , the posterior probabilities are