Page 12 Probability and Conditional Probability

Probability of Events

Definition: The Addition Rule of Probability calculates the probability of either event A or event B occurring.
- Mathematical Formula:
- For events A and B, the formula is given by:
  - P(A ext{ or } B) = P(A) + P(B) - P(A ext{ and } B)
- If events A and B are mutually exclusive (cannot happen at the same time), the formula simplifies to:
  - P(A ext{ or } B) = P(A) + P(B)
Example 1:
- Given:
- P(A) = 0.5
- P(B) = 0.3
- P(A ext{ and } B) = 0.1
- Calculation:
- P(A ext{ or } B) = 0.5 + 0.3 - 0.1 = 0.7
Example 2: Rolling a six-sided die:
- Define events:
- A: Rolling an even number
- B: Rolling a 3
- Calculation for A:
- A has three favorable outcomes (2, 4, 6) out of six total outcomes, so:
  - P(A) = \frac{3}{6} = \frac{1}{2}
- Calculation for B:
- There is one favorable outcome (3), so:
  - P(B) = \frac{1}{6}
- Since you cannot roll both an even number and a 3 at the same time, events A and B are mutually exclusive.
- Thus:
  - P(A ext{ or } B) = P(A) + P(B) = \frac{1}{2} + \frac{1}{6}
  - Convert 1/2 to sixths:
  - \frac{1}{2} = \frac{3}{6}
  - Therefore:
  - P(A ext{ or } B) = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

Definition: Conditional Probability refers to the probability of event B occurring given that event A has occurred.
- Mathematical Formula:
- P(B | A) = \frac{P(A \text{ and } B)}{P(A)}
- Alternatively, it implies:
- P(A \text{ and } B) = P(A) \cdot P(B | A)
Example 3:
- Define events:
- A: Event that a card is a king
- B: Event that a card is a face card
- To find the conditional probability of event B given event A, we would be looking at the probability of drawing a face card conditioned on having drawn a king initially.