Summary of "Probability of Independent and Dependent Events (6.2)"

This video explains the concepts of independent and Dependent Events in Probability, illustrating how to calculate the Probability of combined events in each case.

Main Ideas and Concepts

1. Independent Events

Definition: Two events are independent if the occurrence of one does not affect the Probability of the other.
Example: Rolling a Die and Flipping a Coin.
- Rolling a 6 does not change the Probability of getting heads on a coin flip.
Probability Calculation:
- Formula: P(A and B) = P(A) × P(B)
- Example problem: Probability of rolling a 5 and getting heads.
  - Probability of rolling a 5: 1/6
  - Probability of heads: 1/2
  - Combined Probability: 1/6 × 1/2 = 1/12 ≈ 0.0833

2. Dependent Events

Definition: Two events are dependent if the occurrence of one event affects the Probability of the other.
Example: Drawing Marbles from a box without replacement.
- Box contains 10 marbles: 7 green, 3 blue.
- Drawing a green marble first changes the total marbles and composition for the second draw.
Common Mistake: Using the Independent Events formula for Dependent Events.
Probability Calculation:
- For Dependent Events, the formula adjusts to: P(A and B) = P(A) × P(B | A) where P(B | A) is the Probability of event B occurring after event A has occurred.
- Example 1: Probability of drawing a green marble and then a blue marble without replacement.
  - P(green first) = 7/10
  - After one green marble is removed, 9 marbles remain with 3 blue.
  - P(blue second | green first) = 3/9
  - Combined Probability: 7/10 × 3/9 = 7/30 ≈ 0.233
- Example 2: Probability of drawing two green marbles without replacement.
  - P(green first) = 7/10
  - After one green marble is removed, 6 green marbles remain out of 9 total.
  - P(green second | green first) = 6/9
  - Combined Probability: 7/10 × 6/9 = 7/15 ≈ 0.4667

Recap

Independent Events: Outcome of one event does not affect the other. P(A and B) = P(A) × P(B)
Dependent Events: Outcome of one event does affect the other (common in draws without replacement). P(A and B) = P(A) × P(B | A)

Methodology / Instructions for Calculating Probabilities

For Independent Events:
1. Identify the probabilities of each event.
2. Multiply the probabilities together.
For Dependent Events:
1. Identify the Probability of the first event.
2. Adjust the sample space and favorable outcomes for the second event based on the first event’s outcome.
3. Calculate the conditional Probability of the second event.
4. Multiply the Probability of the first event by the conditional Probability of the second event.