Section 3.2: Conditional Probability and the Multiplication Rule

Introduction to Conditional Probability

Conditional probability is a fundamental concept in probability theory, focusing on the likelihood of an event occurring given that another event has already taken place. This section explores how to calculate such probabilities and introduces the multiplication rule for finding the probability of two events occurring in sequence.

• Conditional Probability: The probability of a second event occurring, given that a first event has already occurred.

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

• Example: P(A | studying) is the probability of getting an A, given that the student studied.

Example: Drawing Cards Without Replacement

• Probability of pulling a Jack from a standard deck:

• Probability of pulling a 10, given a Jack has already been pulled (without replacement):

• This demonstrates that the first event (pulling a Jack) affects the probability of the second event (pulling a 10).

Independent and Dependent Events

Understanding whether events are independent or dependent is crucial for determining the correct probability rules to apply.

• Independent Events: The occurrence of one event does not affect the probability of the other event.

• Dependent Events: The occurrence of one event affects the probability of the other event.

Criteria for Independence:

• Events A and B are independent if or .

• Events are dependent if .

Steps to Determine Independence:

1. Calculate as a stand-alone event.

2. Calculate , the probability of B given A.

3. Compare the two values. If equal, events are independent; if not, they are dependent.

Example: Tossing a Coin Multiple Times

• Probability of getting heads on the next toss is always , regardless of previous outcomes.

• This illustrates independence: past outcomes do not affect future probabilities.

Example: Drawing Two Cards Without Replacement

• Probability first card is a King:

• If first card was a King:

• If first card was not a King:

• Without replacement makes events dependent; with replacement, events are independent.

The Multiplication Rule for Probability

The multiplication rule is used to find the probability that two events, A and B, both occur. The rule differs depending on whether the events are independent or dependent.

• For Dependent Events:

• For Independent Events:

• The rule can be extended to more than two independent events.

Steps for Using the Multiplication Rule:

1. Find the probability the first event occurs.

2. Find the probability the second event occurs, given the first event has occurred.

3. Multiply these two probabilities.

Note: Multiply fractions directly; convert to decimals only at the end to avoid rounding errors.

Example: Probability of Choosing a Jack and a Ten Without Replacement

• Events are dependent.

• This is considered an unusual event (probability ≤ 0.05).

Applications of the Multiplication Rule

Example: Multiple Choice Quiz

• 5 questions, each with 4 choices (1 correct).

• Each question is independent.

• A. Probability first question is correct:

• B. Probability all questions are correct: (unusual)

• C. Probability first question is wrong:

• D. Probability all questions are wrong:

• E. Probability at least one question is correct:

Complement Rule: The probability of at least one success is 1 minus the probability of no successes.

Example: Battery Testing

• 16 batteries tested; 4 failed. Two selected without replacement (dependent events).

• A. Both fail: (unusual)

• B. Both pass:

• C. At least one fails:

Empirical Probability and Contingency Tables

Empirical probability is calculated using observed data, often organized in tables. The following table summarizes student enrollment in math courses by major.

• Empirical Probability Formula:

Examples Using the Table

• A. Probability a student is taking college algebra:

• B. Probability a student is taking college algebra, given nursing major:

• Since , these events are dependent.

• C. Probability a student is a nursing major, given taking algebra:

• D. Probability a student is taking algebra and is a nursing major:

Probability with Given Percentages or Fractions

Example: Standardized Test Submission

• 57.8% of students took a standardized test; 64.4% of those plan to submit their score.

• A. Probability a student took the test and will submit the score:

• B. Probability a student took the test and will not submit the score:

Example: Medical Recovery Probabilities

• Chance of surviving surgery: ; chance of full recovery given survival:

• A. Probability of surviving surgery and making a full recovery:

• B. Probability of surviving surgery and not making a full recovery:

Summary Table: Key Probability Rules

Additional info: An event is considered unusual if its probability is less than or equal to 0.05 (5%).