Question
Download Solution PDFIf A, B and C are arbitrary events, then P (A ∩ B ∩ C) equals to:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is \(P(A) P(B | A)P(C | A∩B)\)
Key Points
- The symbol \(∩\) represents the intersection of events, meaning "and".
- The probability of the intersection of events \(A, B, and C (A ∩ B ∩ C)\) is equal to the probability of A times the conditional probability of B given A times the conditional probability of C given the intersection of A and B.
Additional Information
- The conditional probability of\(B\) given \(A (P (B | A))\) represents the probability of event B occurring given that event A has occurred.
- The conditional probability of C given the intersection of A and \(B (P (C | A ∩ B)) \)represents the probability of event C occurring given that both events A and B have occurred.
- This formula represents the chain rule of probability, which states that the probability of multiple events occurring in a certain order is equal to the product of their individual probabilities.
Important Points
- Understanding the chain rule of probability and the concept of conditional probability is important in various fields, including statistics and computer science.
- The formula \(P (A ∩ B ∩ C) = P (A) P (B | A) P (C | A ∩ B)\) is widely used in decision making, particularly in risk analysis and game theory.
- Careful application of the chain rule is crucial in avoiding incorrect calculations and reaching incorrect conclusions.
Last updated on Jul 8, 2025
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