Lecture Notes: Independent and Dependent Events

Introduction to Event Types

• Independent Events: Events that do not depend on each other.
• Dependent Events: Events that depend on each other.

Example Problem

• Bag of marbles:
  • 8 red marbles
  • 7 blue marbles
  • 6 green marbles
  • 4 yellow marbles
  • Total marbles = 25

Probability Calculations

A. Probability of Selecting a Red Marble

• Formula: Probability = (Number of favorable outcomes) / (Total possible outcomes)
• Calculation:
  • Red marbles = 8
  • Total marbles = 25
  • Probability = 8/25 = 0.32 or 32%

B. Probability of Selecting a Blue Marble then a Green Marble (With Replacement)

• With Replacement: Return the first marble before selecting the second.
• Probability of Blue Marble:
  • Blue marbles = 7
  • Probability = 7/25
• Probability of Green Marble:
  • Green marbles = 6
  • Probability = 6/25
• Combined Probability:
  • (7/25) * (6/25) = 42/625 = 0.0672 or 6.72%
• Independent Events: Blue marble selection does not affect green marble probability.*

C. Probability of Selecting a Yellow Marble then a Red Marble (Without Replacement)

• Without Replacement: Do not return the first marble.
• Probability of Yellow Marble:
  • Yellow marbles = 4
  • Probability = 4/25
• Probability of Red Marble after Yellow Marble:
  • Red marbles = 8
  • New total marbles = 24
  • Probability = 8/24
• Combined Probability:
  • 4/25 * 8/24 = 4/75 ≈ 0.0533 or 5.33%
• Dependent Events: Removing a marble changes total count and affects subsequent probabilities.*

Independent vs Dependent Events

• Independent Events: Event probabilities remain unchanged by previous events
  • Example: With replacement, each event is independent.
• Dependent Events: Probabilities are affected by previous events
  • Example: Without replacement, the event affects total count and subsequent probabilities.

D. Probability of Selecting Two Blue Marbles (With Replacement)

• With Replacement: Independent events.
• Probability of first Blue Marble:
  • 7/25
• Probability of second Blue Marble:
  • 7/25 (same probability)
• Combined Probability:
  • 49/625 = 0.0784 or 7.84%

E. Probability of Selecting Two Green Marbles (Without Replacement)

• Without Replacement: Dependent events.
• Probability of first Green Marble:
  • 6/25
• Probability of second Green Marble:
  • 5/24 (after one is removed)
• Combined Probability:
  • After simplification: 1/20 = 0.05 or 5%

Key Concepts

• Independent Events: Probability of subsequent events does not change.
• Dependent Events: Probability of subsequent events changes due to the first event.
• Replacement vs Without Replacement: Determines whether events are independent or dependent.

Use these notes to understand how to calculate probabilities for independent and dependent events, and the impact of replacement on event dependencies.