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. Jamar Sampson 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. P(X NF)=P(X|F)/P(F)isatruestatement. This reasoning is from gained from getting all probabilities of both above listed events while avoiding counting twice events that have already occurred. P(X NF)isequaltoP(XNF)*P(F). All probabilities must add up to one and not go over. Therefore all listed values equal to 1, backing up the initial statement. (b) Explain why P(F|X)=P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem. Let us prove that P(F(X)) = P(x-F) *P(F) - P(X) using Bayer's Theorem. P(X|F)=P(XnF)/P(F) P(XNF)=P(X|F)*P(F) P(X|F)+P(FnX)/P(X) Since P(F NX)equalsP(X NF) P(X |F)equalsP(X NF)dividedbyP(X). The end result is P(FnX)=P(X|F)*P(F)/P(X)forall. 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). O = Foreign Users I = Domestic Users L = Everyday Users P(O) = 0.7 P(I) = 0.8 P(E) = 0.6 P(X)=P(X|F)*P(F)+P(X|F)*P(F) This means (0.6 * 0.7) + (0.8 * 0.3) = 0.66. Our percentage is 0.66. (b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are from inside the country? P(I|L)=(0.3*0.8)dividedby(0.66). This is also can be represented as (4 / 11). This represents the probability that the users are from inside the country.
snhu PROBLEM 2 This question has 2 parts. Part 1: The drawing below shows a Hasse diagram for a partial order on the set: {A, B, C, D, E, F, G, H, I, J} D G J C H E B I F A Figure 1: A Hasse diagram shows 10 vertices and 8 edges. The vertices, represented by dots, are as follows: vertex J is upward of vertex H; vertex H is upward of vertex I; vertex B is inclined upward to the left of vertex A; vertex C is upward of vertex B; vertex D is inclined upward to the right of vertex C; vertex E is inclined upward to the left of vertex F; vertex G is inclined upward to the right of vertex E. The edges, represented by line segments between the vertices