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Suppose that F and X are events from a common sample space with P(F) # 0 and P(X) #0. Prove that PX) = P(X|F)P(F) + P(X\F)P(F). Hint: Explain why P(X|F)P(F) = P(Xn F) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1.

Suppose that F and X are events from a common sample space with P(F) # 0 and P(X) #0. Prove that PX) = P(X|F)P(F) + P(X\F)P(F). Hint: Explain why P(X|F)P(F) = P(Xn F) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1.

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
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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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Discrete Mathematics I need help with this 2 part problem Part 1 Part 2
o.tex - TeXstudio label tiny 2 of 14 Part 1: Suppose that Fand X are events from a common sample space with P(F) #0 and P(X) #0. (a) Prove that PX) = P(X|F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F) = P(X nF) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1. (b) Explain why PLAX) = PX\F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem. Part 2: A website reports that 70% of its users are from outside a certain country. Out of their users from outside the country, 60% of them log on every day. Out of their users from inside the country. 80% of them log on every day. (a) What percent of all uses log on every day? Hint: Use the equation from Part 1 (a). (b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are from inside the country?
Transcribed Image Text:o.tex - TeXstudio label tiny 2 of 14 Part 1: Suppose that Fand X are events from a common sample space with P(F) #0 and P(X) #0. (a) Prove that PX) = P(X|F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F) = P(X nF) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1. (b) Explain why PLAX) = PX\F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem. Part 2: A website reports that 70% of its users are from outside a certain country. Out of their users from outside the country, 60% of them log on every day. Out of their users from inside the country. 80% of them log on every day. (a) What percent of all uses log on every day? Hint: Use the equation from Part 1 (a). (b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are from inside the country?
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Given that F and X are events from a common sample space with P(F) ≠ 0 and P(X) ≠ 0.

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