Tasks 07-04 - Conditional Probability

Section 07: Probability & Statistics

Problem 1: Basic Conditional Probability (x)

Given \(P(A) = 0.6\), \(P(B) = 0.5\), and \(P(A \cap B) = 0.3\):

  1. Find \(P(A|B)\)
  2. Find \(P(B|A)\)
  3. Are A and B independent?

Problem 2: Multiplication Rule (x)

A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement.

  1. Find \(P(\text{both red})\)
  2. Find \(P(\text{first red, second blue})\)
  3. Find \(P(\text{different colors})\)

Problem 3: Tree Diagrams (xx)

A company has two production lines: - Line A produces 60% of items, with 5% defect rate - Line B produces 40% of items, with 8% defect rate

  1. Draw a tree diagram
  2. Find \(P(\text{defective})\)
  3. Find \(P(\text{Line A and defective})\)
  4. Given an item is defective, find \(P(\text{from Line A})\)

Problem 4: Law of Total Probability (xx)

A store has three suppliers: - Supplier X: 50% of stock, 2% return rate - Supplier Y: 30% of stock, 4% return rate - Supplier Z: 20% of stock, 5% return rate

  1. What is the overall return rate?
  2. A returned item is selected. What’s the probability it came from Supplier Y?

Problem 5: Sequential Selection (xx)

From a group of 8 men and 5 women, 3 people are selected randomly (without replacement).

  1. Find \(P(\text{all women})\)
  2. Find \(P(\text{all men})\)
  3. Find \(P(\text{at least one woman})\)
  4. Find \(P(\text{exactly 2 men})\)

Problem 6: Independence Testing (xxx)

A survey of 400 employees collected data on job satisfaction (Satisfied/Not Satisfied) and work arrangement (Remote/Office):

Remote Office Total
Satisfied 90 150 240
Not Satisfied 60 100 160
Total 150 250 400
  1. Find \(P(\text{Satisfied})\)
  2. Find \(P(\text{Satisfied}|\text{Remote})\)
  3. Find \(P(\text{Satisfied}|\text{Office})\)
  4. Are satisfaction and work arrangement independent?

Problem 7: Exam-Style Problem (xxx)

At a university, 70% of students pass the statistics exam. Of those who pass, 80% studied more than 10 hours. Of those who fail, 30% studied more than 10 hours.

  1. Draw a complete tree diagram with all probabilities
  2. Find \(P(\text{studied more than 10 hours})\)
  3. A student studied more than 10 hours. What’s the probability they passed?
  4. Are “passing” and “studying more than 10 hours” independent?