Probability and Conditional Probability

Conditional Probability

• Conditional probability occurs when the probability of an event depends on another event occurring first.

• It reduces the event space, increasing the probability of the desired outcome.

• Notation: P(B|A) represents the probability of event B given that event A has occurred.

• Formula: P(B|A) = \frac{P(A \cap B)}{P(A)}, where P(A) > 0

• Also: P(A \cap B) = P(A|B)P(B)

Applying Conditional Probability

• In probability problems, "knowing that" or "given" indicates conditional probability, which changes the sample space.

Multiplication of Probabilities for Dependent Events

• For events "without replacement," individual probabilities must be adjusted each time.

• "And then" indicates multiplication of probabilities.

• Example: Probability of drawing an aqua ball (A) and then a black ball (B) without replacement:
  P(A, \text{ then } B) = P(A) \cdot P(B|A)

• Extended multiplication: Probability of multiple successive events.
  P(B, B, B) = P(B) \cdot P(B|B) \cdot P(B|B, B)

Probability Tree Diagrams

• Multiply along the branches to find the probability of A and then B: P(A \cap B) = P(A) \times P(B|A).

• To find the probability that A occurs, sum the probabilities of all branches where A occurs.

Independence of Events

• Events are independent if one has no effect on the other.

• For independent events A and B: P(A \cap B) = P(A)P(B).

• Relative frequencies from data can estimate conditional probabilities and indicate potential independence.