In all parts of this problem, assume that we are using fair,regular dice (six-sided with values 1, 2, 3, 4, 5, 6 appearingequally likely). Furthermore, assume that all dice rolls aremutually independent events.
(a) [4 pts] You roll two dice and look at the sum of the facesthat come up. What is the expected value of this sum? Express youranswer as a real number.
(b) [7 pts] Assuming that the two dice are independent,calculate the variance of their sum. Express your answer as a realnumber.
(c) [7 pts] You repeatedly roll two fair dice and look at thesum. What is the probability that you will roll a sum of 4 beforeyou roll a sum of 7? Express your answer as a real number.
(d) [7 pts] What is the expected number of rolls until you get asum of 4 or a sum of 7? (For example, if you get 7 on the firstroll, the number of rolls is 1.) Express your answer as a realnumber.
(e) [7 pts] You roll 10 dice. Using the Chernoff Bound, give anupper bound for the probability that 8 or more of them rolled a 1or a 2? You don’t need to calculate the value with a calculator(since you do not have one), but please write it in simplestterms.
