Problems: 1. [5 points] Suppose our sample space is Ω = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 6}, B = {3, 4}, C = {1, 4, 5}. Find (a) A ∪ B (b) A ∩ B (c) (B ∩ C)c 2. [15 points] The following problems involve applying Kolmogorov’s axioms for probability laws, and properties that follow from Kolmogorov’s axioms. (a) Out of the students in this class, 60% can attend my office hours if I hold them on Monday, 75% can attend them on Friday, and 40% can attend either day. What is the probability that a randomly selected student will not be able to attend my office hours? (b) There is a 50% chance it rains on Saturday, a 60% chance it rains on Sunday, and a 70% chance it rains on Saturday OR Sunday. What is the percent chance it rains on Saturday AND Sunday? (c) Consider the following statements: • the chance it rains on Saturday is 50% 1 • the chance it rains on Sunday is 70% • the chance it rains on both Saturday AND Sunday is 15% Are these statements consistent? In other words, is it possible for all three statements to be true? Justify your answer. (Hint: compute P(A ∪ U) and verify if it satisfies the the axioms.) 3. [5 points] The pieces of candy you are eating come in three flavors: orange, grape, and cherry. When you reach into the bag, the probability you pull out an orange or grape piece is 0.7, while the probability you pull out a grape or cherry piece is 0.9. What is the probability that you pull out an orange or cherry piece? (Hint: Let O, G, and C denote the events that you pull out an orange piece, grape piece, and cherry piece, respectively. You will find the normalization property 1 = P(O) + P(G) + P(C) helpful.) 4. [5 points] Suppose A is a real number chosen on [0, 1] according to the uniform law. Using the value of A, define the quadratic polynomial f(x) = Ax2 + 3x + 4. What is the probability that the roots of f(x) are real? (Hint: The roots of f(x) will be real if and only if the discriminant of the quadratic polynomial is nonnegative.) 5. [15 points] Suppose that a point (x, y) is chosen from the unit square S = [0, 1] × [0, 1] using the uniform probability law — the probability that (x, y) is in a subset A of S is equal to the area of A: P((x, y) ∈ A) = area(A) for all A ⊂ S. (a) What is the probability that x + y < 1/3 ? (b) What is the probability that x + y < 3/2 ? (c) For any real number u, define F(u) := P(x + y ≤ u). Find an expression for F(u) in terms of u. (Hint: for different ranges of u, your expression may change. Just break your answer into a few different cases depending on u.)