For the following problem, clearly describe the sample space and the random vari

Question

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.

(1) First assume n = 1 (i.e., you only roll one six-sided die).

(a) (1 point) Describe the sample space for this experiment.

(b) (1 point) Describe a random variable which maps an outcome of this experiment to the winnings you receive.

(c) (1 point) Compute the expected value of this random variable.

(2) Now assume n = 6 (i.e., you roll six six-sided dice).

(a) (1 point) Describe the sample space for this experiment (you don’t 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 six 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.

Explanation / Answer

Solution:

1

a)

when n=1, our sample space will be 6 i.e outcome of a single dice if rolled alone and those outcomes will be {1, 2, 3, 4, 5, 6}

b)

random variable for winning we receive will be 1 or 6 (x= 1 or 6).

c)

Expected value of random variable,

P(x= 1 or x=6)= 1/6+1/6= 2/6.

1/6 is a probability of getting a 1 and 1/6 for 6, so combining both will give us 2/6.

2)

a)

If n=6, our sample space= 6^6= 46656

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