#### Applied Probability

What is the Law of Large numbers? The following exercise explains it. If you consider a random sample of size n, as

2. Design an experiment to explain the Law of Large Numbers using Coin Toss (use the website on Question #1 or StatCrunch). Write down the instructions to explain the Law of Large Numbers (In other words, you are designing an activity to explain the Law of Large Numbers)

Use a standard 52-card deck for this activity. Use appropriate notations in your answers. (a) Compute the probability of the event E ="drawing a king".

a) Proof P(AUBUC)= P(A)+P(B)+P(C)âˆ’P(An B)-P(ANC)-P(BNC) +P(AnBnC) b) Let A, B, and C are independent, P(A)=0.3, P(B)= 0.4, P(C)=0.5 Find P(AU BUC)

(5) Considering the NBA playoffs, 16 teams will participate in the playoffs, 8 teams will reach the conference semifinals, 4 teams will reach the conference finals, and 2 teams will play in the NBA finals. For a total of 16 teams, how many possible combinations are there? (It will be a large number, you might consider only using factorials).

A system consists of four components connected as shown in the following diagram: Assume A,B,C, and D function independently. If the probabilities that A, B, C, and D fail are 0.10, 0.05, 0.010 and 0.20, respecvtively, what is the probability that the system functions?

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1. A space S and three of its subsets are given by S = {1, 3, 5, 7, 9, 11}, A = {1,3,5}, B = {7, 9, 11}, and C = {1, 3, 9, 11}. Find An BnC, Aºn B, A-C, and (A - B) UB.

Prove the second part of DeMorgan's Law, i.e., show that (AUB) = Aºn Bº.

Consider an experiment consisting of rolling a die twice. The outcome of this experiment is an ordered pair whose first element is the first value rolled and whose second element is the second value rolled. (a) Find the sample space. (b) Find the set A representing the event that the value on the first roll is greater than or equal to the value on the second roll. (c) Find the set B corresponding to the event that the first roll is a six. (d) Let C correspond to the event that the first valued rolled and the second value rolled differ by two. Find An C.

Let A, B, and C' be events in an event space. Find expressions for the following: (a) Exactly one of the three events occurs. (b) Exactly two of the events occurs. (c) Two or more of the events occur. (d) None of the events occur.

Consider an experiment consisting of randomly selecting a real number r between 0 and 1. The sample space is then the interval (0, 1), since the outcome can be any number in this interval. (a) Find the set A representing the event that r is between 0.3 and 0.4. (b) Find the set B representing the event that r is between 0.35 and 0.6. (c) Find the set C' representing the event that r is either less than 0.25 or greater than 0.75. (d) Find An B.

A system is composed of five components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcomes of the ex- periment be given by all vectors X1, X2, X3, X4, X5), where x¿ is 1 if component i is working and 0 if component i is not working. (a) How many outcomes are in the sample space of this experiment? (b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components 1, 3, and 5 are all working. Let W be the event that the system will work. Specify all of the outcomes in W. (c) Let A be the event that components 4 and 5 have both failed. How many outcomes are in the event A. (d) Write out all outcomes in the event An W.

Consider two particles of masses m₁ and m₂ in orbit about their common center of mass. Let F be defined as the position vector of the particle of mass m, relative to the center of mass. Part A: Derive the equation of motion for m₁ and show that it can be written in the form. \ddot{\overline{\Gamma_{1}}}+\frac{G m_{2}{ }^{3}}{\left(m_{1}+m_{2}\right)^{2} r_{1}^{3}} \overline{r_{1}}=\overline{0} Part B: Derive the equation of motion for m₂ and demonstrate that it is similar, that is, \ddot{r_{2}}+\frac{G m_{1}^{3}}{\left(m_{1}+m_{2}\right)^{2} r_{2}^{3}} \bar{r}_{2}=\overline{0} Part C: Finally, prove that the relative equation that results from (a) and (b) is \frac{\ddot{r}}{r_{12}}+G\left(m_{1}+m_{2}\right) \frac{\bar{r}_{12}}{r_{12}^{3}}=\overline{0}