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madreJ [45]
3 years ago
14

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.
Mathematics
1 answer:
katrin [286]3 years ago
8 0

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.

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arlik [135]
The practice of tacking 9/10 of a cent on the end of a gas price goes back to when gas cost only pennies per gallon and was a tax imposed by state and federal governments. Gas stations added the fraction of a cent on the end of the price instead of rounding up the price.
8 0
3 years ago
If the subspace of all solutions of Ax 0 has a basis consisting of vectors and if A is a ​matrix, what is the rank of​ A
Nuetrik [128]

Question: If the subspace of all solutions of

Ax = 0

has a basis consisting of vectors and if A is a ​matrix, what is the rank of​ A.

Note: The rank of A can only be determined if the dimension of the matrix A is given, and the number of vectors is known. Here in this question, neither the dimension, nor the number of vectors is given.

Assume: The number of vectors is 3, and the dimension is 5 × 8.

Answer:

The rank of the matrix A is 5.

Step-by-step explanation:

In the standard basis of the linear transformation:

f : R^8 → R^5, x↦Ax

the matrix A is a representation.

and the dimension of kernel of A, written as dim(kerA) is 3.

By the rank-nullity theorem, rank of matrix A is equal to the subtraction of the dimension of the kernel of A from the dimension of R^8.

That is:

rank(A) = dim(R^8) - dim(kerA)

= 8 - 3

= 5

4 0
3 years ago
Bob decided to give up a full-time salary of $45000 a year to go to school for 4 years. The total cost of going to school will n
den301095 [7]

Answer:

Step-by-step explanation:

Semester Costs = 8*2858 =                     22864

Books / semester= 8 * 391 =                        <u>3128</u>

Total                                                            25992

If he wants to repay all this in six years the answer would be

45000 + 25992/6 = 45000 + 4332 = 49332

3 0
3 years ago
Which equation has a solution of n = 4?
Allushta [10]

Answer:

D. 7n = 28

Step-by-step explanation:

Let's substitute 4 for n in each equation to check if it the resulting equation is correct:

6+n=24\\ 6+4=24\\ 10\neq 24\rightarrow \text{Incorrect!}

5n=54\\5(4)=54\\20\neq54\rightarrow\text{Incorrect!}

12-n=16\\12-4=16\\8\neq16\rightarrow\text{Incorrect!}

7n=28\\7(4)=28\\28=28\rightarrow\text{Correct!}

Therefore the answer is D

4 0
2 years ago
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<span> 2.632 x 10^4  ÷  2 x 10-7 =

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