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In the following questions A, B and C are events. (a) (5 points) Prove Boole's inequality for three events: P(AU BUC) ≤ P(A) + P(B) + P(C). (b) (5 points) Prove that if P(C) > 0 and A, B are independent, then: P(ABC) = P(C) · P(An B|C) (c) (5 points) Suppose that P(A|C) ≤ P(B|C) and P(A|CC) ≤ P(B|Cº). Show that P(A) ≤ P(B). (d) (5 points) Show that if A, B are independent, and B, C are independent, it does not necessarily follow that A and C are independent. You can either prove this or provide a counterexample.

In the following questions A, B and C are events. (a) (5 points) Prove Boole's inequality for three events: P(AU BUC) ≤ P(A) + P(B) + P(C). (b) (5 points) Prove that if P(C) > 0 and A, B are independent, then: P(ABC) = P(C) · P(An B|C) (c) (5 points) Suppose that P(A|C) ≤ P(B|C) and P(A|CC) ≤ P(B|Cº). Show that P(A) ≤ P(B). (d) (5 points) Show that if A, B are independent, and B, C are independent, it does not necessarily follow that A and C are independent. You can either prove this or provide a counterexample.

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter11: Data Analysis And Probability
Section11.9: Independent And Dependent Events
Problem 1E
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help me with abcd please. please handwrite if possible. please don't use AI tools to answer

In the following questions A, B and C are events. (a) (5 points) Prove Boole's inequality for three events: P(AU BUC) ≤ P(A) + P(B) + P(C). (b) (5 points) Prove that if P(C) > 0 and A, B are independent, then: P(ABC) = P(C) · P(An B|C) (c) (5 points) Suppose that P(A|C) ≤ P(B|C) and P(A|CC) ≤ P(B|Cº). Show that P(A) ≤ P(B). (d) (5 points) Show that if A, B are independent, and B, C are independent, it does not necessarily follow that A and C are independent. You can either prove this or provide a counterexample.
Transcribed Image Text:In the following questions A, B and C are events. (a) (5 points) Prove Boole's inequality for three events: P(AU BUC) ≤ P(A) + P(B) + P(C). (b) (5 points) Prove that if P(C) > 0 and A, B are independent, then: P(ABC) = P(C) · P(An B|C) (c) (5 points) Suppose that P(A|C) ≤ P(B|C) and P(A|CC) ≤ P(B|Cº). Show that P(A) ≤ P(B). (d) (5 points) Show that if A, B are independent, and B, C are independent, it does not necessarily follow that A and C are independent. You can either prove this or provide a counterexample.
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