Conditional probability is playing a crucial role in statistics, decision-making, and risk assessment. It focuses on evaluating the probability of one event occurring, given that another event has already taken place. Understanding this interplay between events is vital for a deep comprehension of probability theory.

Basic Concept

• Definition: Conditional probability is the chance of event A happening given that event B has already happened, denoted as P(A|B).
• Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, with P(B) > 0.

Understanding Events

• Interdependence: Knowing how one event affects another is crucial.
• Sample Spaces: All possible outcomes in a probability scenario.
• Tree Diagrams: Useful for visualizing probabilities in complex situations.

Example Scenarios

A. Dice and Coin Toss:

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• Events: A = "heads on coin toss", B = "even number on die".
• Calculations:
  • $P(B) = \frac{3}{6}$ (3 even numbers on die).
  • $P(A \text{ and } B) = \frac{1}{2} \times \frac{3}{6} = \frac{1}{4}$ (independent events).
  • $P(A|B) = \frac{(1/4)}{(3/6)} = \frac{1}{2}$.

B. Card Draw from a Deck:

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• Events: A = "drawing a king", B = "drawing a face card".
• Calculations:
  • $P(B) = \frac{12}{52}$ (12 face cards in deck).
  • $P(A \text{ and } B) = \frac{4}{52}$ (4 kings, all face cards).
  • $P(A|B) = \frac{(4/52)}{(12/52)} = \frac{1}{3}$.