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3.8. Let A, B, and C be events relating to the experiment of rolling a pair of dice. a. If P(AC) > P(B|C) and P(A/C) > P(B|C) either prove that P(A) > P(B) or give a counterexample by defining events A, B, and C for which that relationship not true. b. If P(A|C) > P(A|C) and P(B|C) > P(B|C) either prove that P(AB|C) > P(AB|C) or give a counterexample by defining events A, B, and C for which that relationship is not true. Hint: Let C be the event that the sum of a pair of dice is 10; let A be the event that the first die lands on 6: let B be the event that the second die lands on 6

3.8. Let A, B, and C be events relating to the experiment of rolling a pair of dice. a. If P(AC) > P(B|C) and P(A/C) > P(B|C) either prove that P(A) > P(B) or give a counterexample by defining events A, B, and C for which that relationship not true. b. If P(A|C) > P(A|C) and P(B|C) > P(B|C) either prove that P(AB|C) > P(AB|C) or give a counterexample by defining events A, B, and C for which that relationship is not true. Hint: Let C be the event that the sum of a pair of dice is 10; let A be the event that the first die lands on 6: let B be the event that the second die lands on 6

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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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How can I prove letter a and b from question 3.8.?

**Probability Problem on Conditional Events** **Problem 3.8: Understanding Conditional Probabilities with Dice** *Context*: Consider the experiment of rolling a pair of dice, and let \( A \), \( B \), and \( C \) be events in this context. **Part (a):** Given: - \( P(A|C) > P(B|C) \) - \( P(A|C^c) > P(B|C^c) \) Task: - Either prove that \( P(A) > P(B) \) - Or present a counterexample by defining events \( A \), \( B \), and \( C \) where this relationship does not hold. **Part (b):** Given: - \( P(A|C) > P(A|C^c) \) - \( P(B|C) > P(B|C^c) \) Task: - Either prove that \( P(AB|C) > P(AB|C^c) \) - Or provide a counterexample with definitions for events \( A \), \( B \), and \( C \) where this relationship is false. **Hint for Both Parts**: - Define \( C \) as the event where the sum of the pair of dice is 10. - Let \( A \) be the event where the first die lands on 6. - Let \( B \) be the event where the second die lands on 6. Use these definitions to explore and either support the given relationships or find conditions under which they do not hold.
Transcribed Image Text:**Probability Problem on Conditional Events** **Problem 3.8: Understanding Conditional Probabilities with Dice** *Context*: Consider the experiment of rolling a pair of dice, and let \( A \), \( B \), and \( C \) be events in this context. **Part (a):** Given: - \( P(A|C) > P(B|C) \) - \( P(A|C^c) > P(B|C^c) \) Task: - Either prove that \( P(A) > P(B) \) - Or present a counterexample by defining events \( A \), \( B \), and \( C \) where this relationship does not hold. **Part (b):** Given: - \( P(A|C) > P(A|C^c) \) - \( P(B|C) > P(B|C^c) \) Task: - Either prove that \( P(AB|C) > P(AB|C^c) \) - Or provide a counterexample with definitions for events \( A \), \( B \), and \( C \) where this relationship is false. **Hint for Both Parts**: - Define \( C \) as the event where the sum of the pair of dice is 10. - Let \( A \) be the event where the first die lands on 6. - Let \( B \) be the event where the second die lands on 6. Use these definitions to explore and either support the given relationships or find conditions under which they do not hold.
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