Introduction to Statistics - Pragel Name:________________________ Project 2 November 7, 2022 Please turn in this project to the Brightspace Dropbox by 9:30am (the start of class) on Monday, November 28th. Late projects will be penalized and may not be accepted. This will be graded out of 30 points. Your project will consist of the completed table below as well as the answers to the questions on the next page. For this project, you will be looking at expected value. Suppose that a casino is offering a new game: If you roll two dice and the difference between the dice is 2, you win $10. Otherwise you lose $3. Roll two dice with your partner to complete the following table. Each person should have their table completely filled in. Record the values of both dice, the difference, and the win/loss amount. Then compute the total won/lost, as well as the average amount won/lost per game for your sample of 20 experiments. [9 points]

Trial Values of both dice Difference (largest-smallest) Win/Loss amount Example (3,6) 6 - 3 = 3 $ - 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Trial Values of both dice Difference (largest-smallest) Win/Loss amount 15 16 17 18 19 20 Total won/lost: Average win/loss per game (round to 2 decimal places):

Questions: 1.) [2 points] Based on your simulation, do you think that the casino should offer this game? Explain why or why not.

2.) Another way to approach this problem is to use theoretical probability. a. [3 points] Write out the sample space for rolling two dice. There should be a total of 36 possible outcomes.

b. [4 points] Using your sample space from part a, circle all of the outcomes where the player wins (the difference of the dice was two.). What is the probability that the player wins? What is the probability the player loses (casino wins)?

c. [4 points] Using part b, compute the expected value of this game to the player using the theoretical probability. Round your answer to two decimal places.

d. [3 points] Interpret the expected value of this game to the player. Using this interpretation, do you think the casino should continue offering this game? Explain why or why not.

3.) [3 point] Does your empirical average win/loss per game match your theoretical expected value? Should these values match? Explain why or why not.

4.) [2 points] How could the casino change the winning amount to make this into a fair game (where neither the player nor the casino have an expected value advantage?)