Step-by-step explanation:
(a) You win $5 if you roll a six, $1 if you roll an odd number, and $0 if you roll a 2 or a 4, and you pay $1.50 for every roll. The expected value is the sum of each outcome multiplied by its probability.
E = (5.00)(1/6) + (1.00)(3/6) + (0)(2/6) + (-1.50)(1)
E = -0.167
You are expected to lose on average about $0.17 per roll, which means Alex has the advantage.
(b) The probability of rolling a 2 or 4 on a fair die is 2/6 or 1/3. The probability of this happening five times is:
P = (1/3)⁵
P = 1/243
P ≈ 0.41%
There is approximately a 0.4% probability that a fair die will roll a 2 or 4 five times.
(c) The confidence interval for a proportion is:
CI = p ± ME
ME = CV × SE
The margin of error is the critical value times the standard error.
The critical value for 95% confidence is z = 1.960.
The standard error for a proportion is:
SE = √(pq/n)
Given p = 1/6, q = 5/6, and n = 100:
SE = √((1/6) (5/6) / 100)
SE = 0.037
So the confidence interval is:
CI = 1/6 ± (1.960) (0.037)
CI = 0.167 ± 0.073
0.094 < p < 0.240
Since the observed proportion of 0.08 is outside of this interval, we can conclude with 95% confidence that the die is not fair.
(d) Under the current game rules and die probabilities, the expected value is:
E = (5.00)(0.08) + (1.00)(0.33) + (0)(0.59) + (-1.50)(1)
E = -0.77
To make the game fairer, but to still give Alex the advantage so she can make money for her fundraiser, we need to change the rules of the game so that the expected value is less negative.
One simple way to do this is to pay players $2.00 if they roll a 2.
Now the expected value is:
E = (5.00)(0.08) + (1.00)(0.33) + (2.00)(0.29) + (0)(0.30) + (-1.50)(1)
E = -0.19
Now instead of expecting to lose on average $0.77 per roll, players can expect to lose on average $0.19 per roll. This means they have a better chance of winning money, but Alex still has the advantage.
