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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2011 Math 411 — Problem Set 1 Issued: 09.02 Due: 09.09 1.1. Two dice are rolled. What is the probability that (a) the two numbers will differ by 1 or less and (b) the maximum of the two numbers will be 5 or larger? 1.2. Suppose Ω = {a, b, c}, P ({a, b}) = 0.7, and P ({b, c}) = 0.6. Compute the probabilities of {a}, {b}, and {c}. 1.3. Given two events A and B with P (A) = 0.4, and P (B) = 0.7, what are the maximum and minimum possible values for P (A ∩ B)? 1.4. A family has 3 children, each of whom is a boy or a girl with probability 1/2. Let A = “there is at most 1 girl” B = “the family has children of both sexes.” (a) Are A and B independent? (b) Are A and B independent if the family has 4 children? 1.5. Roll two dice. Let A = “the sum is even” and B = “the sum is divisible by 3,” that is B ∈ {3, 6, 9, 12}. Are A and B independent? 1.6. Suppose we roll three tetrahedral dice that have 1, 2, 3, and 4 on their four sides. Find the distribution of the sum of the three numbers.