Section 5.3: Conditional Probability

Definition and Calculation of Conditional Probability

Conditional probability quantifies the likelihood of an event occurring given that another event has already occurred. It is a foundational concept in probability theory and statistics.

• Conditional Probability of event A given event B is denoted as P(A|B).

• The formula is:

• P(A|B) is read as "the probability of event A, given event B."

• The vertical slash "|" represents the word "given."

• Of the times B occurs, P(A|B) is the proportion of times that A also occurs.

Visualizing Conditional Probability

Venn diagrams are useful for visualizing conditional probabilities. The denominator in the conditional probability formula always refers to the "given" event.

• For P(A|B), consider only the region where B occurs; within that, the proportion where A also occurs is the conditional probability.

• P(A|B) is not necessarily equal to P(B|A).

Example 1: Rolling a Die

• Question: What is the probability of obtaining a 6 when you roll a die, assuming that you got an even number?

• Solution: Define event A = {6}, event B = {2, 4, 6}.

• Interpretation: Of the times B occurs (even numbers), the proportion of times A occurs (rolling a 6) is 1 out of 3.

Example 2: Vehicles Sample

• Question: In a sample of 40 vehicles, 15 are red, 6 are trucks, and 2 are both. What is the probability that a randomly selected red vehicle is a truck?

• Solution:

Example 3: Tax Audit and Contingency Tables

Contingency tables are used to organize data for two categorical variables, such as income level and audit status.

Frequencies are in thousands. For example, 1,233 represents 1,233,000 tax forms that reported income under $200,000 and were audited.

Conditional Probabilities from the Table

Each cell probability is the frequency divided by the total number of returns (e.g., ).

• Example Calculation: Probability of being audited given income at least $1,000,000:

• Let A = audited=yes, B = income ≥ $1,000,000

Multiplication Rule for Joint Probability

The multiplication rule allows us to find the probability that two events both occur.

• For any events A and B:

• Alternatively:

Example 4: Double Faults in Tennis

• Novak Djokovic made 62% of his first serves; he faulted on 38% of first serves.

• Given a first serve fault, he faulted on the second serve 4.5% of the time.

• Let F1 = fault on first serve, F2 = fault on second serve.

• Probability of double fault:

• Djokovic makes a double fault in 1.7% of all his service games.

Sampling With and Without Replacement

Sampling methods affect the probabilities of selecting items from a population.

• Sampling without replacement: Once a subject is selected, it cannot be selected again.

• Sampling with replacement: Subjects can be selected more than once.

Independence of Events

Two events are independent if the occurrence of one does not affect the probability of the other.

• Events A and B are independent if or equivalently .

• If A and B are independent, then .

Checking for Independence

• To determine if A and B are independent, check if any of the following hold:

• If one is true, the others are also true, and A and B are independent.

Example 5: The Triple Blood Test for Down Syndrome

A study of 5,282 women aged 35 or over analyzed the accuracy of the Triple Blood Test for Down syndrome.

• Positive test (POS): Test states the condition is present.

• Negative test (NEG): Test states the condition is not present.

• False Positive: Test states the condition is present, but it is actually absent.

• False Negative: Test states the condition is absent, but it is actually present.

Probability Calculations

• Probability of a positive test for a randomly chosen woman:

• Are POS and D (Down Syndrome) independent?

• Since , the events are dependent.

Additional info: The notes also introduce the concepts of false positives and false negatives, which are important in diagnostic testing and medical statistics.