Probability (7 points)

For the following problem, clearly describe the sample space and the random variables you use. Be sure to justify where you get your expected values from. Consider playing a game where you roll n fair six-sided dice. For every 1 or 6 you roll you win $30, for rolling any other number you lose $3n (where n is the total number of dice rolled).

(2) Now assume n = 7 (i.e., you roll 7 six-sided dice).

(a) (1 point) Describe the sample space for this experiment (you dont need to list the elements but describe what is contained in it).

(b) (1 point) Describe a random variable which maps an outcome of this experiment to the winnings you receive. (Hint: Express your random variable as the sum of seven random variables.)

(c) (1 point) Compute the expected value of this random variable using the linearity of expectation. Based on this would you play this game?

(3) (1 point) What is the largest value of n for which you would still want to play the game? Justify your answer.