Consider the Monty Hall problem.Verify the results using by writing a computer program that estimates the probabilities of winning for different strategies by simulating it.

1. First, write a code that randomly sets the prize behind one of three doors and you also randomly select one of the doors. You win if the the door you selected has the prize (Here, we are simulating ’stick to the initial door’ strategy). Repeat this experiment 100 times and compute the average number of wins.

2. Next, try simulating the switching strategy. Find the door the host will open and change your initial door with the door not opened by the host. Also repeat this experiment 100 times and compute the average number of wins. If you did everything right, the first code should yield the probability of winning as ≈ 1/3 and the second code should yield ≈ 2/3. You can use any programming language you want (MATLAB, Python etc.)

A newly discovered amino acid can exist in two conformations; A, and B. Conformer A is 2.4 kJ/mol higher in energy than conformer B. Calculate the ratio of the two conformers at 20 o C.

True or False (write word in blank)

_________1. If A and B are finite sets then |A x B| = |A| · |B|

_________2. |A È B| = |A| + |B| only if A Ç B = Æ

________ 3. The multiplication principle finds the total number of branches of decision tree.     

________ 4. There are 256 binary strings of length 8.

________ 5. There are 160 binary strings of length 8 that begin with 00 or have 1 as the 8^th digit.

________ 6. There are 7 ways to arrange 1’s and 0’s in a binary string of length 8 without having 2 zeros in a row.                                            

________7. If a man has 2 suits, 5 shirts, and 10 ties, he has 17 different outfits.

________8. There are 100,000 possible social security numbers if the first 3 numbers are 404.

________9. There are 17,576,000 possible 6 character license plates if the first 3 characters are letters and the last 3 characters are numbers.

_______10. The addition principle is used to count the number of possible outcomes for disjoint sets

Consider the non linear ODE:

(dx/dt) = -y = f(x,y)

(dy/dt) = x^2-x = g(x,y)

(a). Compute all critical points (b) Derive the Jacobian matrix (c). Find the Jacobians for each critical point (d). Find the eigenvalues for each Jacobian matrix (e). Find the linearized solutions in the neighborhood of each critical point (f) Classify each critical point and discuss their stability (g) Sketch the local solution trajectories in the neighborhood of each critical point

1. Calculate gcd(181451, 186623).

2. For integers a, b, and c, if a | bc, then either a | b or a | c.

solve

x^2y^''-4xy^'+6y=lnx^2

y(0)=0, y^'(0)=1

Suppose a small cannonball weighing 20 pounds is shot vertically upward, with an initial velocity v0 = 340 ft/s. The answer to the question "How high does the cannonball go?" depends on whether we take air resistance into account. If air resistance is ignored and the positive direction is upward, then a model for the state of the cannonball is given by d2s/dt2 = −g (equation (12) of Section 1.3). Since ds/dt = v(t) the last differential equation is the same as dv/dt = −g, where we take g = 32 ft/s2. If air resistance is incorporated into the model, it stands to reason that the maximum height attained by the cannonball must be less than if air resistance is ignored. (a) Assume air resistance is proportional to instantaneous velocity. If the positive direction is upward, a model for the state of the cannonball is given by m dv dt = −mg − kv, where m is the mass of the cannonball and k > 0 is a constant of proportionality. Suppose k = 0.0025 and find the velocity v(t) of the cannonball at time t.

Please Abstract Algebra and Group actions

A benzene molecule is six carbon atoms arranged on the vertices of a regular hexagon. How many chemical compounds can be created by attaching either NH₂, COOH, or OH to each one of the carbon atoms in a benzene molecule? Two molecules are considered the same if one molecule can be rotated or reflected (acted on by D₆) to get the other molecule.

3. Consider the IVP:

dy =ty^1/3; y(0)=0,t≥0. dt

Both y(t) = 0, (the equilibrium solution) and y(t) = ?(1/3t^2?)^3/2 are solutions to this IVP.

1. (a) Show that the trivial solution satisfies the IVP by first verifying that it satisfies the initial condition and then

   verifying that it satisfies the differential equation.

2. (b) Show that the other solution satisfies the IVP again by first verifying it satisfies the initial condition and then verifying that it satisfies the differential equation.

3. (c) Explain in your own words why having these two solutions does not violate the existence and uniqueness theorem we discussed in class.

A tank contains100 gal of water and 50 oz of water and 50 oz of salt. water containing salt concentration of (1/4)(1 + (1/2)sint) oz/gal flows into the tank at a rate of 2gal/min, and the mixture in the tank flows out at the same rate.

a) find the amount of salt in the tank at any time

b) plot the solution for a time period long enough so that you see the ultimate behavior of the graph

c) the long-time behavior of solution is an oscillation about a certain constant level. what is the level? what is the amplitude of the oscillation?

Exercise 3. (22 pts.) (Rabbits vs. Sheep) a)For each of the following ”rabbits vs. sheep” problems with x, y ≥ 0. Find the fixed points, classify them (type and stability), and find the eigenvalues and eigenvectors. Then sketch (by hand) a plausible phase portrait indicating nullclines, all relevant trajectories, and indicate all the different basins of attraction. Finally interpret the behavior of the population of the rabbits and sheep for the different systems (compare the three systems)

• x˙ = x(3 − 2x − y), ˙y = y(2 − x − y)

What is the exact point of intersection of the curves f(x)=x^5-x^3 and g(x)=sin(x), when x>0? This is for a Calculus II Maple Lab, for the volume of revolution for x>/=0 about x=pi.

Consider the initial value problem

mx′′+cx′+kx=F(t),   x(0)=0,   x′(0)=0

modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=100cos(8t) Newtons.

Solve the initial value problem.

x(t)=

Determine the long-term behavior of the system (steady periodic solution). Is limt→∞x(t)=0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t.

For very large positive values of t,

x(t)≈xsp(t)=

My friend picks five card from the 52-card standard deck. He shows me only two of the cards: A♥ and K♦. What is the probability that my friend scored a full house?