[Solved] SOLVED:Homework 1

1. Consider a fair four-sided die, with sides 1, 2, 3, and 4, that is rolled twice. For example, 1,4 would indicate 1 was rolled rst and then 4 was rolled second.(a) Write down the possible outcomes, i.e., the sample space. (b) List the outcomes in the following events: Event A: The number 4 came up zero times. Event B: The number 4 came up exactly one time. Event C: The sum of the two rolls is odd. Event D: The second roll was one more than the rst roll. (c) True or False: i. Events A and B are disjoint. ii. Events B and C are disjoint. iii. Event B is a subset of event C. iv. Event D is a subset of event C. (d) Compute the probability P [AB]. (e) Compute the probability P [BC]. (f) Compute the probability P [C D]. 2. Walpole Exercise 2.14. Copied here: If S = {0,1,2,3,4,5,6,7,8,9}and A = {0,2,4,6,8}, B = {1,3,5,7,9}, C = {2,3,4,5}, and D = {1,6,7}, list the elements of the sets corresponding to the following events: (a) AC; (b) AB; (c) C0; (d) (C0D)B; (e) (S C)0; (f) AC D0. 3. Among 110 students in a probability class, on a three-question exam, 29 get question 1 incorrect, 39 get question 2 incorrect, and 22 get question 3 incorrect. Further, 19 students get both questions 1 and 2 incorrect, 13 get both questions 2 and 3 incorrect, and 7 get both questions 1 and 3 incorrect, and 6 get all three questions incorrect. Find the probability that a randomly selected student: (a) Gets question 1 incorrect but gets question 2 correct. (b) Gets questions 2 and 3 incorrect but gets question 1 correct. (c) Gets all three questions correct. 4. A tablet manufacturer produces tablets that may or may not have the following specs: A=32 GB memory; B=64 GB memory; C=color screen; D=black & white screen. You are given that C and D are disjoint and C D = S, where S is the sample space. You are given that A and B are disjoint, and there are other options for memory size. We are give the following probabilities: P [A] = 0.7 and P [B] = 0.1, P [C] = 0.8 and P [BC] = 0.01. (a) Draw a Venn diagram for A,B,C,D, and S. Note that in a Venn diagram, the size of a set doesnt need to be proportional to the actual event probability, however, events that are disjoint should not overlap and events that are not disjoint should overlap. Events that form a partitions in a Venn diagram should have union that covers the sample space. (b) What is the probability P [AB]? (c) What is the probability P [[AB]c]? (d) What is the probability P [BD]? (e) What is the probability P [D]?