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# 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 ﬁrst 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 ﬁrst 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 [A∪B]. (e) Compute the probability P [B∪C]. (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) A∪C; (b) A∩B; (c) C0; (d) (C0∩D)∪B; (e) (S ∩C)0; (f) A∩C ∩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 [B∩C] = 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 doesn’t 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 [A∪B]? (c) What is the probability P [[A∪B]c]? (d) What is the probability P [B∩D]? (e) What is the probability P [D]?

(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 [B∩C] = 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 doesn’t 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 [A∪B]? (c) What is the probability P [[A∪B]c]? (d) What is the probability P [B∩D]? (e) What is the probability P [D]?

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