Question

In all parts of this problem, assume that we are using fair, regular dice (six-sided with values 1, 2, 3, 4, 5, 6 appearing equally likely). Furthermore, assume that all dice rolls are mutually independent events.

(a) [4 pts] You roll two dice and look at the sum of the faces that come up. What is the expected value of this sum? Express your answer 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 real number.

(c) [7 pts] You repeatedly roll two fair dice and look at the sum. What is the probability that you will roll a sum of 4 before you 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 a sum of 4 or a sum of 7? (For example, if you get 7 on the first roll, the number of rolls is 1.) Express your answer as a real number.

(e) [7 pts] You roll 10 dice. Using the Chernoff Bound, give an upper bound for the probability that 8 or more of them rolled a 1 or a 2? You don’t need to calculate the value with a calculator (since you do not have one), but please write it in simplest terms.

Answer 1

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