snhu MAT 230 EXAM TWO This document is proprietary to Southern New Hampshire University. It and the problems within may not be posted on any non-SNHU website. Sharmaine Manlansing 1
snhu 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. To prove this: We utilize conditional probability for events F and X, given that P(F) > 0 and P(X) > 0. The conditional probability for X given F is: P(X|F) = P(F) P(XnF) For the complement of F, the conditional probability is: P(X|F)= P(XnF) P(F) To prove the right-hand side: P(X|F).P(F)+P(XF).P(F) Rewrite: = P(X|F).P(F)+P(XF).P(F) _P(XOF).P(F)+P(XnF).P(F) P(F)+P(F) = P(Xn(FUF)), where FnF ¥ 0 = P(Xn), where= (X) This leads to the conclusion: P(X)=P(X|F).P(F)+P(X|F).P(F) (b) Explain why P(F|X)=P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem. P(X|F)×P(F)=P(XNF) P(X|F)×P(F)=P(X|F)×P(F) Thus, we conclude: P(F(X))=P(X|F)×P(F)+P(X|F)×P(F) 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. snhu (a) What percent of all users log on every day? Hint: Use the equation from Part 1 (a). A: Out of the country B:Out of the country (every day) C: In the country (everyday) P(A) =. 70 P(C/A) =. 60 P(C/B) =. 80 P(C/A) =. 80 P(B) = P(A) = 1 -. 70 -. 30 P(C)=P(C|A)*P(A)+P(C|?)*P(A) (.60 * . 70) + (.80 * . 30) .42 +.24 P(C) =. 66 (b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are from inside the country? P(B/C)=P(B)*P(C|B)/P(A)*P(C|A)+P(B)*P(C/B) .30 * . 80/.66 P(B|C)=4/11