Answer to Question #297565 in Discrete Mathematics for sdmnh

Question

This question has 2 parts. Part 1: Suppose that F and X are events from a common sample space with P(F) 6= 0 and P(X) 6= 0. (a) Prove that P(X) = P(X|F)P(F) + P(X|F¯)P(F¯). Hint: Explain why P(X|F)P(F) = P(X ∩ 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. (b) Explain why P(F|X) = 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 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?