Directions : Type your solutions into this document and be sure to show all steps for arriving at your solution Just giving a final number may not receive full credit . PROBLEM 1 This question has 2 parts. Part 1 Suppose that F and X are events from a common sample space with P(F) ne0 and P(X) ne0 . (a) Prove that P(X)=P(X|F)P(F)+P(X| overline F )P( overline F ) . Hint: Explain why P(X|F)P(F)=; P(X cap 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?

Directions: Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit. PROBLEM 1 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. (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 n 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(FIX) = 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