Please type the response, I have a problem reading alot of handwritings. **Problem 1**This question has 2 parts.**Part 1**: Suppose that \( F \) and \( X \) are events from a common sample space with \( P(F) \neq 0 \) and \( P(X) \neq 0 \).(a) Prove that \( P(X) = P(X|F)P(F) + P(X|F^c)P(F^c) \). Hint: Explain why \( P(X|F) = \frac{P(X \cap F)}{P(F)} \) from the definition of conditional probability, and then use that with the logic from the proof of Theorem 1.1.(b) Explain why \( P(F|X) = \frac{P(X|F)P(F)}{P(X)} \) 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 also log on every day. Hint: Use the equation from Part 1(a).(a) What percent of all users log on every day?(b) Using Bayes' Theorem, out of users who log on every day, what is the probability that they are from inside the country?

Name: Please type the response, I have a problem reading alot of handwriting
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: Please type the response, I have a problem reading alot of handwritings. **Problem 1**This question has 2 parts.**Part 1**: Suppose that \( F \) and \( X \) are events from a common sample space with \( P(F) \neq 0 \) and \( P(X) \neq 0 \).(a) Prove