Multiplication Rule for Dependent Events and Conditional Probability

Multiplication Rule: Dependent Events

Multiplication Rule for Dependent Events

The multiplication rule is a fundamental concept in probability that allows us to find the probability of two events both occurring. When events are dependent, the outcome of one event affects the probability of the other.

• Independent Events: The probability of both events A and B occurring is the product of their individual probabilities.

• Dependent Events: The probability of both events A and B occurring is the probability of A multiplied by the probability of B given that A has occurred.

Formula for Independent Events:

• $P(A \text{ and } B) = P(A) \times P(B)$

Formula for Dependent Events:

• $P(A \text{ and } B) = P(A) \times P(B|A)$

Example: Drawing Marbles

• A bag contains 7 red and 6 blue marbles. Find the probability of drawing a red marble and then a blue marble:

• Independent: If the first marble is replaced, the events are independent.

• Dependent: If the first marble is not replaced, the events are dependent.

Event	Independent Events	Dependent Events
Drawing a red marble from the bag, replacing it, then drawing a blue marble	$P(A \text{ and } B) = P(A) \times P(B) = \frac{7}{13} \times \frac{6}{13}$
Drawing and keeping a blue marble from the bag, then drawing a red marble		$P(A \text{ and } B) = P(A) \times P(B|A) = \frac{6}{13} \times \frac{7}{12}$

Example: What is the probability that a card player draws two aces from a standard deck of 52 cards if they keep the first card after drawing it?

• For dependent events (no replacement): $P(\text{first ace}) = \frac{4}{52}$, $P(\text{second ace}|\text{first ace}) = \frac{3}{51}$

• For independent events (with replacement): $P(\text{first ace}) = \frac{4}{52}$, $P(\text{second ace}) = \frac{4}{52}$

Conditional Probability

Definition and Formula

Conditional probability is the probability of event B occurring given that event A has already occurred. It is denoted as $P(B|A)$.

• Formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

This formula can be rearranged from the multiplication rule for dependent events.

Example: Survey of Students

• A statistics class has 43 students: 20 math majors, 15 science majors, 8 business majors. 4 students have a double major in math & science.

• Find the probability that a randomly selected student has a science major, given that the student has a math major.

• Given event (A): math major

• Other event (B): science major

• $P(A) = \frac{20}{43}$

• $P(A \text{ and } B) = \frac{4}{43}$

• $P(B|A) = \frac{P(A \text{ and } B)}{P(A)} = \frac{4/43}{20/43} = \frac{4}{20} = 0.2$

Example: Pets in a Town

• 15% of people in a town own both a cat and a dog. 40% of residents have a dog. What is the probability that someone in the town owns a cat, given they have a dog?

• $P(\text{cat and dog}) = 0.15$

• $P(\text{dog}) = 0.40$

• $P(\text{cat}|\text{dog}) = \frac{0.15}{0.40} = 0.375$

Applications of the Multiplication Rule

Book Selection Example

Probability rules for dependent events are commonly used in real-world scenarios such as random selection without replacement.

• A library selects two monthly book club reads by randomly choosing two books from a list of 100 adult novels. 62 books are fiction, 38 are nonfiction. What is the probability of choosing two nonfiction books?

• First book: $P(\text{nonfiction}) = \frac{38}{100}$

• Second book (without replacement): $P(\text{nonfiction}|\text{first nonfiction}) = \frac{37}{99}$

• Total probability: $P(\text{both nonfiction}) = \frac{38}{100} \times \frac{37}{99}$

Job Application Example

• A company is hiring software developers and data analysts. Out of 14 applicants, 6 applied for the software developer position, 8 for the data analyst position. 4 of the software developer applicants have a degree in computer science, and 2 of the analyst applicants also have a degree in computer science. What is the probability that a randomly selected candidate with a degree in computer science applied for the software developer position?

• Given event (A): degree in CS

• Other event (B): applied for software developer

• $P(A) = \frac{6}{14}$

• $P(A \text{ and } B) = \frac{4}{14}$

• $P(B|A) = \frac{4/14}{6/14} = \frac{4}{6} = 0.667$

Summary Table: Multiplication Rule for Events

Type of Events	Formula	Example
Independent	$P(A \text{ and } B) = P(A) \times P(B)$	Drawing two cards with replacement
Dependent	$P(A \text{ and } B) = P(A) \times P(B|A)$	Drawing two cards without replacement
Conditional Probability	$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$	Probability of science major given math major

Additional info: These notes cover material relevant to Chapter 5 (Probability in Our Daily Lives) and Chapter 9 (Statistical Inference: Significance Tests About Hypotheses), focusing on the multiplication rule for dependent events and conditional probability, with practical examples and applications.