4. Suppose we are working with a circular dart board with 6 concentric regions labelled 1 (outermost) to 6 (disc at the center). Assume that the dart board has radius r and that a dart thrown at the board always hits the board. It is easy to see that the probability of scoring i points on a single throw is (a) Show that P(scoring i points) = Area of region i Area of the dart board P(scoring i points) = (7-i)²-(6-1)² 6² Note that this probability is independent of the radius of the dartboard, why? (b) Show that function constructed above satisfies the probability axioms. (c) Show that the probability function constructed above is an increasing function of i. 5. Suppose P is a probability function on the pair (sample space, sigma algebra): (S,B). Using only the three defining properties of P, and for any A, B € B, show that: (a) P(A) = 1- P(A). (b) P(AUB) = P(A) + P(B) - P(ANB). (c) If A C B, then P(A) ≤ P(B).

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.

2. CHEATING ON AN EXAM A professor has learned that three students in her class of 20 will cheat on the exam. She decides to focus her attention on four randomly chosen students during the exam. a) Find the probability that she finds at least one of the students cheating if she focuses on four randomly chosen students? b) Find the probability that she finds at least one of the students cheating if she focuses on six randomly chosen students?

5. Probability distribution of a sum of two integer random variables. Examples foridentically distributed summands with Poisson, geometric, uniform distributions.

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.

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)

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.

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.

FORUM DESCRIPTION Part (a): Create a discrete probability distribution using the generated data from the following simulator: Anderson, D. Bag of M&M simulator. New Jersey Factory. Click on the simulator to scramble the colors of the M&Ms. Next, add the image of your generated results to the following MS Word document: Discrete Probability Distributions.docx. Use the table in this document to record the frequency of each color. Then, compute the relative frequency for each color and include the results in the table. Part (b): Compute the mean and standard deviation (using StatCrunch or the formulas) for the discrete random variable given in the table from part (a). Include your results in the MS Word document. Part (c): Add a screenshot of the entire completed MS Word document to the discussion board as part of your discussion response (do NOT upload the MS Word document). Part (d): Comment on your findings in your initial response and respond to at least 2 of your classmate's findings. Grading Information

3. MPG RATING OF CARS Suppose that the miles-per-gallon rating of passenger cars is a normally distributed random variable with a mean of 33.8 mpg and standard deviation of 3.5 mpg. a) What is the probability that a randomly selected passenger car gets at least 40 mpg. b) What is the probability that a randomly selected passenger car gets at most 30 mpg. c) What is the probability that a randomly selected passenger car gets between 30 and 40 mpg. d) An automobile manufacturer wants to build a new passenger car with an mpg rating that improves upon 99% of existing cars. What is the minimum mpg that would achieve this goal? DICK