1 a Calculate the probability generating function for a rand

1. (a) Calculate the probability generating function for a random variable whose value is determined by the face shown after a roll of a fair six-sided die. Simplify this using the formula for the sum of a finite geometric series.

(b) Use your result from (a) to calculate the probability generating function for a random variable whose value is the sum of the face values of three independent rolls of a fair six-sided die.

2. In playing a certain game, your ability scores are determined by six independent rolls of three dice as in problem 1. After each set of six rolls, you are given the choice of keeping your scores or starting over.

(a) How many times should you expect to start over in order to get a set of ability scores with at least two scores that are 18?

(b) What is the probability that it will take longer than this number of trials to get a set of ability scores with at least two scores that are 18?

(c) Repeat (a) and (b) for a set of ability scores with at least two scores that are 16 or larger and no scores that are 6 or lower.

Solution