please explain the multiplication rule as it applies to the Rules of Probability in at least 400 words

Use a LaPlace transform to solve

d^2x/dt^2+dx/dt+dy/dt=0

d^2y/dt^2+dy/dt-4dy/dt=0

x(0)=1,x'(0)=0

y(0)=-1,y'(0)=5

Q5. During peak hours, customers arrive at the cashier check-out queue in a local supermarket according to a Poisson process with an average rate of 120 customers per hour. There are four cashiers serving the check-out, and they provide identical service to customers. The time to service any customer by any of the cashier is exponentially distributed with a mean of 20 seconds. If all the cashiers are busy, customers join in a single queue on arrival.

(a) Compute the probability that an arriving customer have to wait in the queue for an available cashier;
(b) Suppose the supermarket opens at 10am, and Peter arrives at the cashier check-out at 10.02am. What assumptions are needed to approximate the probability that Peter do not have to queue.

Pls explain with workings. Thxs

1. Compute the product in the given ring.

a) (16)(12) in Z₂₄

b) (-4)(11) in Z₅

c) (2,4)(4,7) in Z₅ x Z₉

2. Describe all units in the given ring.

a) Z₇

b) Z₈

c) Z x Z x Q

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that (p − 1)2 = 1 (mod p)

2. Is the above true for any number (not necessarily prime)?

3. Show that the equation a 2 − 1 = 0, has only two solutions mod p.

4. Consider (p − 1)!. Show that (p − 1)! = −1 (mod p) Remark: Think about what are the values of inverses of 1, 2, . . . , p − 2.

List ONLY positive relationships between two variables. (Please list as many as possible, the more the better.)

Consider an ecological niche with three species A, B and C. Their dynamics are given by their population rates shown below

dA/dt=A+B-C … (2a)

dB/dt=-A+2B … (2b)

dC/dt=C-A-B … (2c)

Initially, the niche is in equilibrium with the population of C=30. Obtain the population densities of the rest of participants in the niche at its equilibrium. Starting from the point (A=8,B=4,C=10), obtain their individual populations over time t. What if the initial starting point is perturbed to (A=7,B=4,C=10)? Would the system emerge in the same way?

1. Determine an inverse of a modulo m for a = 6 and m = 11.  This is equivalent to answering the question “_______ is the unique inverse of 6 (mod 11) that is non-negative and < 11.”  Show your work following the steps.

1. Beside the inverse you identified in part a), identify two other inverses of 6 (mod 11).  

Hint:  All of these inverses are congruent to each other mod 11.

3. Although the congruence can be solved using any of the inverses you identified in parts a) and b), use the unique inverse of 6 (mod 11) that you identified in part a), that is, the non-negative inverse of 6 (mod 11) that is < 11, to solve the following congruence.  Show your work.  6x ≡ 8(mod 11).

4. Determine if the congruence 6x ≡ 11(mod 8) has a solution.

   If there is a solution, identify a value for x.  If there is no solution, explain why not.

(a) Give a definition of a closed set.

(b) Show, directly from the definition, that a union of finitely many closed sets is closed.

(c) Give an example of a countable collection of closed intervals In such that ∪ n=1 to ∞ I_n is open (make sure to prove it).

MATLAB question

The range of a projectile launched at velocity V and angle q

is R=2 V² sin(q) cos(q)

1. Find the range if V=20m/s and q=30 degrees (5 pts)
2. If the uncertainty in the launch angle q = 30 +/- 2 degrees, what is the uncertainty of the range

What should the accuracy of the launch angle have be to keep the uncertainty of the range to within 5%.

Find the positive root of the equation x^3-×-11 using inspection method.

1.- Prove that the set of irrational numbers is uncountable by using the Nested Intervals Property.

2.- Apply the definition of convergent sequence, Ratio Test or Squeeze Theorem to prove that a given sequence converges.

3.- Use the Divergence Criterion for Sub-sequences to prove that a given sequence does not converge.

Subject: Real Analysis

Prove Euler's Formula for graph inductively.

1. How many 12-bit strings (that is, bit strings of length 14) start with the sub-string 011?

2. You break your piggy bank to discover lots of pennies and nickels.

You start arranging them in rows of 6 coins.

How many coins would you need to make all possible rows of 6 coins (not necessarily with equal numbers

of pennies and nickels)?

3. How many shortest lattice paths start at (4, 4) and end at (13, 13)?

4. What is the coefficient of x⁵y² in the expansion of (x + y)⁷?