Your friend Alex is planning a fundraising game night to raise money for a local children's hospital. She invites a few of your friends over to test out some of the games using fake Monopoly money.

For her first game, you will roll a six sided die and you will win $5 if you roll a six, $1 if you roll an odd number, and $0 if you roll a 2 or a 4, you will also pay $1.50 for every roll. Before you decide to play you do a few calculations. You have watched several people play this game and decided to collect some data.

(25 points) Assuming that it is a fair six-‐sided die, what are your expected winnings if you play Alex’s game? Who has the advantage in this game? Show your work and explain your result in a complete sentence.
(25 points) You’ve been watching a few people play this game and have observed that in the last five rolls none
f the players have won any money; that is, all rolls have been 2’s and 4’s. If Alex was using a fair die what is the probability of rolling only 2’s and 4’s five times in a row? Show your work and convey your results in a complete sentence.
You begin to suspect that Alex’s die is unfair, and decide to collect data regarding the results from several players. You would like to determine if the data you have observed is plausible assuming the die is fair.

c) (25 points) Of the 100 rolls you observed, a 6 is rolled only eight times. Based on this data, construct and interpret a 95% confidence interval for the proportion of rolls that are a 6 using Alex’s die. Do you believe that Alex’s die was a fair die? Justify your response with complete sentences, based on the confidence interval you constructed.

The following table displays the all the data from the 100 rolls you observed.

Outcome Frequency
1 10
2 29
3 11
4 30
5 12
6 8
You show Alex your findings, and she is embarrassed. She would like to develop a dice game that is fairer for the guests at her fundraiser.

d) (25 points) Using the empirical data above, give advice to Alex on how she should set up a game using her dice where she can still make money for her fundraiser but it might be more enticing for the guests to play. Give her an example of a game that she can use. Justify your recommendation with an expected value calculation and explain its significance to the situation in complete sentences.