2.4 Conditional probability

• The probabilities assigned to various events depend on

  • what is known about the experimental situation
  • when the assignment is made.

• Subsequent to the initial assignment, partial information relevant to the outcome of the experiment may become available.

• Such information may cause us to revise some of our probability assignments.

• For a particular event $A$ ,

  • we have used $P\left(A\right)$ to represent the probability, assigned to $A$ ;
  • we now think of $P\left(A\right)$ as the original, or unconditional probability, of the event $A$ .

• In this section, we examine how the information “an event $B$ has occurred” affects the probability assigned to $A$ .

• For example, $A$ might refer to an individual having a particular disease in the presence of certain symptoms.

  • If a blood test is performed on the individual and the result is negative $\left(B=\text{negative blood test}\right)$,
  • then the probability of having the disease will change
    • it should decrease, but not usually to zero, since blood tests are not infallible

• We will use the notation $P\left(A\mid B\right)$ to represent the conditional probability of $A$ given that the event $B$ has occurred.

  • $B$ is the “conditioning event.”

Example

As an example,

• consider the event $A$ that
  • a randomly selected student at your university obtained all desired classes during the previous term’s registration cycle.
• Presumably $P\left(A\right)$ is not very large.
• However, suppose the selected student is an athlete who gets special registration priority (the event $B$ ).
• Then $P\left(A\mid B\right)$ should be substantially larger than $P\left(A\right)$,
  • although perhaps still not close to 1 .

EX 2.24

• In Equation (2.2), the conditional probability is expressed as a ratio of unconditional probabilities:
  • The numerator is the probability of the intersection of the two events,
  • whereas the denominator is the probability of the conditioning event $B$.
• A Venn diagram illuminates this relationship (Figure 2.8).

Figure 2.8 Motivating the definition of conditional probability

• Given that $B$ has occurred, the relevant sample space
  • is no longer $\mathcal{S}$
  • but consists of outcomes in $B$;
• $A$ has occurred if and only if
  • one of the outcomes in the intersection occurred,
• so the conditional probability of $A$ given $B$ is proportional to $P\left(A\cap B\right)$ .
• The proportionality constant $1/P\left(B\right)$ is used to ensure that
  • the probability $P\left(B\mid B\right)$ of the new sample space $B$ equals 1 .

2.4.1 The Definition of Conditional Probability

2.4.2 The Multiplication Rule for union of two sets

2.4.3 Bayes’ Theorem

EXERCISES Section 2.4 (45-69)