Question

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 $8 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 $2 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.

a) 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.

b) You’ve been watching a few people play this game and have observed that in the last six rolls none of 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 six 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) Of the 100 rolls you observed, a 6 is rolled only four 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	33
3	11
4	30
5	12
6	4

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) 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.

Answer 1

a)

The probability of rolling a 6 is 1/6 and the corresponding winnings are $8.

The probability of rolling an odd number (1, 3, or 5) is 3/6=1/2 and the corresponding winning is $1.

The probability of rolling a 2 or a 4 is 2/6=1/3 and the corresponding winnings are $0.

Considering the $2 charge per roll, the expected winnings are:

Thus, a loss of $0.17 is expected per roll of a fair die.

b)

The probability of rolling a 2 or a 4 is 2/6=1/3. So, the probability of rolling a 2 or a 4 6 times in a row is

Thus, the probability of rolling a 2 or a 4 using a fair die is 0.0014.

c)

Given a sample mean of 4 out of 100 = 0.04, we would like to construct a 95% confidence interval for the population mean.

A 95% confidence interval corresponds to a probability range of 0.025 to 0.975. The corresponding z-score is -/+1.96. So, the corresponding confidence interval will be:

So, per 100 rolls, the 95% confidence interval is between 2.04 and 5.96 rolls of 6.

For a fair die, the expected number is 100/6=16.67 rolls of 6. As this is not in the above interval of (2.04, 5.96), we conclude that Alex's die was not a fair die.

d)

The easy way of enticing the guests and still making money is to increase the prize money on 6 from $8 to $37.

The probability of rolling a 6 is 0.04 and the corresponding winnings are $37.

The probability of rolling an odd number (1, 3, or 5) is 0.10+0.11+0.12=0.33 and the corresponding winning is $1.

The probability of rolling a 2 or a 4 is 0.33+0.30=0.63 and the corresponding winnings are $0.

Considering the $2 charge per roll, the expected winnings will now be:

Given the situation, an increase in prize money is justified to entice the people to play the game, make some profit for fundraising, and avoid making an undue profit.