Lecture Notes: Independent and Dependent Events

Introduction

• Independent Events: Events that do not depend on each other.
• Dependent Events: Events that rely on the outcome of another event.

Example Problem

Scenario: A bag contains:

• 8 red marbles
• 7 blue marbles
• 6 green marbles
• 4 yellow marbles

Total Marbles

• Total = 25 marbles (8 + 7 + 6 + 4)

Calculating Probabilities

Part A: Probability of Selecting a Red Marble

• Formula: Number of favorable outcomes / Total outcomes
• Probability: ( \frac{8}{25} \approx 32% )

Part B: Probability of Selecting a Blue Marble & Then a Green Marble (With Replacement)

• Blue Marble: ( \frac{7}{25} )
• Green Marble: ( \frac{6}{25} )
• Combined Probability: ( \frac{42}{625} \approx 6.72% )
• Type: Independent events (the second event does not depend on the first)

Part C: Probability of Selecting a Yellow Marble & Then a Red Marble (Without Replacement)

• Yellow Marble: ( \frac{4}{25} )
• Red Marble: ( \frac{8}{24} )
• Combined Probability: ( \frac{4}{75} \approx 5.3% )
• Type: Dependent events (probability changes because of no replacement)

Understanding Independent vs. Dependent Events

• Independent Events: Probability of second event remains unchanged by the first event.
• Dependent Events: Probability of second event is affected by the first event (usually due to no replacement).

Part D: Probability of Selecting Two Blue Marbles (With Replacement)

• First Blue Marble: ( \frac{7}{25} )
• Second Blue Marble: ( \frac{7}{25} )
• Combined Probability: ( \frac{49}{625} \approx 7.84% )
• Type: Independent events

Part E: Probability of Selecting Two Green Marbles (Without Replacement)

• First Green Marble: ( \frac{6}{25} )
• Second Green Marble: ( \frac{5}{24} )
• Combined Probability: ( \frac{1}{20} \approx 5% )
• Type: Dependent events

Key Takeaways

• With Replacement: Typically results in independent events.
• Without Replacement: Typically results in dependent events.