Problem 3) Show that for any three events A, B, and C with P(C) > 0, P(A U B |C) = P(A| C) + P(B|C) - P(AN B| C)
Problem 3) Show that for any three events A, B, and C with P(C) > 0, P(A U B |C) = P(A| C) + P(B|C) - P(AN B| C)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem 3)** Show that for any three events \(A\), \(B\), and \(C\) with \(P(C) > 0\),
\[ P(A \cup B \mid C) = P(A \mid C) + P(B \mid C) - P(A \cap B \mid C) \]
This formula represents the conditional probability of the union of two events \(A\) and \(B\) given another event \(C\). The left-hand side of the equation, \(P(A \cup B \mid C)\), denotes the probability that either event \(A\) occurs or event \(B\) occurs, given that event \(C\) has occurred.
The right-hand side consists of three terms:
- \(P(A \mid C)\): The probability that event \(A\) occurs given that event \(C\) has occurred.
- \(P(B \mid C)\): The probability that event \(B\) occurs given that event \(C\) has occurred.
- \(P(A \cap B \mid C)\): The probability that both events \(A\) and \(B\) occur given that event \(C\) has occurred.
Therefore, the formula is derived from the principle of inclusion-exclusion in probability, adapted for conditional probabilities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fcf4b0b-d82c-43e8-86bb-da9864dd2a82%2F5e571929-4ce7-4665-b075-32d0cf89ec30%2F39uhtv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3)** Show that for any three events \(A\), \(B\), and \(C\) with \(P(C) > 0\),
\[ P(A \cup B \mid C) = P(A \mid C) + P(B \mid C) - P(A \cap B \mid C) \]
This formula represents the conditional probability of the union of two events \(A\) and \(B\) given another event \(C\). The left-hand side of the equation, \(P(A \cup B \mid C)\), denotes the probability that either event \(A\) occurs or event \(B\) occurs, given that event \(C\) has occurred.
The right-hand side consists of three terms:
- \(P(A \mid C)\): The probability that event \(A\) occurs given that event \(C\) has occurred.
- \(P(B \mid C)\): The probability that event \(B\) occurs given that event \(C\) has occurred.
- \(P(A \cap B \mid C)\): The probability that both events \(A\) and \(B\) occur given that event \(C\) has occurred.
Therefore, the formula is derived from the principle of inclusion-exclusion in probability, adapted for conditional probabilities.
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