Why Conditional Probability?

In many real situations, probability must be evaluated given additional information.

For example:

• What is the probability a student passes an exam given that they attended all lectures?
• What is the probability a machine fails given that it has already operated for two years?

Once information is known, the sample space effectively changes. Conditional probability provides the mathematical framework for updating probabilities under new information.

Definition of Conditional Probability

The probability of event A given that event B has already occurred is called conditional probability.

The formula is:

P(A given B) = P(A and B) / P(B)

where P(B) is greater than zero.

This formula tells us that once event B occurs, the probability of A is determined by the portion of B’s outcomes that also belong to A.

Interpretation of Conditional Probability

Conditional probability can be interpreted as restricting attention to a smaller sample space.

Instead of considering all outcomes, we now consider only outcomes where event B occurs.

Within this restricted set, we examine how often event A occurs.

Example