e with P(F) #0 and P(X) # 0. Explain why P(X\F)P(F) = ional probability, and then use %3D of stating Theorem 4.2.1 Bayes' rtain country. Out of their users r users from inside the country, uation from Part 1 (a). hat is the probability that they

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Please type the response, I have a problem reading alot of handwritings. 

PROBLEM
This question has 2 parts.
Part 1: Suppose that F and X are events from a common sample space with P(F) 0 and P(X) 0.
P(X\F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F)
(a) Prove that P(X)
P(XnF) 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.
||
%3D
(b) Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes'
Theorem.
%3D
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 users 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:PROBLEM This question has 2 parts. Part 1: Suppose that F and X are events from a common sample space with P(F) 0 and P(X) 0. P(X\F)P(F) + P(X|F)P(F). Hint: Explain why P(X|F)P(F) (a) Prove that P(X) P(XnF) 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. || %3D (b) Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes' Theorem. %3D 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 users 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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