Problem 2: You have applied to two universities X and Y for graduate admission. There is a .25 chance that you are going to get into X and .20 chance that you are going to get into Y. Keep in mind that admission into the universities is independent events.

a)    What is the probability that you are going to get into both universities?

b)     What is the probability that you are going to get into at least one university?

c)    What is the probability that you are going to get into any one of the university?

d)    Are getting into those universities mutually exclusive?

In a simple undirected graph H the sum of vertex degrees is 60. What is the smallest possible number of vertices in this graph? What is the largest possible number of vertices in the graph?

• What do you know about how math and historical developments are related? Share any information you already have about the topic.
• Are there any questions you still have?
• Do you believe that the teaching and learning of math has changed?
• What are some of your first conjectures of how math affected the growth of civilization?

Assume we are flipping 1000 fair coins. What can you say about the probability that you will get at least 3/5 of them to be heads? Compare the 3 bounds (Markov, Chebyshev, Chernoff), and try plugging in some leargevalue of n to see how muchs they differ!

You need a loan of $160,000 to buy a home. Calculate your monthly payments and total closing costs for each choice below.

Choice 1: 30 year fixed rate at 5% with closing costs if $2300 and no points.

Choice 2: 30 year fixed rate at 4.5% with closing costs of $2300 and 5 points.

What is the monthly payment for choice one?

What is the monthly payment for choice two?

what is the total closing cost for choice one?

what is the total closing cost for choice two?

why might choice one be better?

why might choice two be better?

The next three questions are based on the following information.

To study the weight accuracy of a 50lb fertilizer bag, 12 samples of 12 bags of fertilizer in each sample were taken and the results are as follows.

1.

The overall average weight of a bag of fertilizer is ________ pound, and the average range is ____________ pound. (Please round to two decimal points and include no units.)

2.

The upper control limit for a 99.7% control chart for the mean is ________ pound, and the lower control limit is __________ pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)

3.  

The upper control limit for a 99.7% control chart for the range is_________ pound, and the lower control limit is _________ pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)

Explain Egyptian arithmetic.

Use the classification of groups with six elements to show that A(4) has no subgroup with 6 elements. [ Hint: check that the product of any two elements of A(4) of order 2 has order 2]

Let A∈R^{n× n} be a non-symmetric matrix.

Prove that |λ₁| is real, provided that |λ₁|>|λ₂|≥|λ₃|≥...≥|λ_n| where λ_i , i= 1,...,n are the eigenvalues of A, while others can be real or not real.

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is injective, but not surjective.

(b) Suppose g : R\{−1} → R\{a} is a function such that g(x) = x−1, where a ∈ R. Determine x+1

a, show that g is bijective and determine its inverse function.

Please review all of the course materials regarding Financial Markets: Savings and Investment Vehicles before you begin this assignment.  You will also need to reference the Tax Cuts and Jobs Act of 2017.

The questions below also rely on the following assumptions:

• You are 30 years old and your employer sponsors a 401(k) plan with a 4% employer match.  
• You earn $100,000 of gross wage income. This income is expected to stay constant over the next three years.
• At the start of every year you decide to invest 4% of your salary into your 401(k).
• Your expected return on your investments is 5% per year.
• You file your taxes as a single filer and you are in the 24% tax bracket.
• The long term capital gains tax is 15%.

A. Calculate the total amount of funds that you expect to be in your 401(k) at the end of three years. Explain your answer.

B. At the end of the third year, you decide to withdraw $15,000 from your 401(k) to pay for some home improvements. Calculate how much tax, if any, you will owe on this withdrawal. Explain your answer.

C. During this same three year period you also invested in a Roth IRA. At the end of this three year period, your Roth IRA had cumulative contributions of $15,000 and earnings or gains of $5,000.

Suppose you decided to fund your home improvements by withdrawing from your Roth IRA instead of your 401(k). Calculate how much tax, if any, would you owe in this withdrawal. Explain your answer.

Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

4e^-x2 sin(x) = x² − x + 1

The neighborhood of a vertex in a graph consists of the vertex itself, together with all vertices that are connected to it by an edge. Each graph has a variable xi associated with the i-th vertex, and the vertex has a known value that is equal to the sum of the variables for all neighborhood vertices. Start with a graph with 5 vertices forming a pentagon, with edges joining vertices 1 and 2, 2 and 3, 3 and 4, 4 and 5, and 5 and 1. Then draw an edge joining vertices 2 and 4, and an edge joining vertices 2 and 5. The known values at vertices 1 through 5 are, respectively, 2, 1, −1, 3, and 5.

(a) Find the augmented matrix for the system of equations satisfied by x₁, x₂, x₃, x₄, x_5.