Problem 1:
Stopping rule in "Deal or No Deal" game show is a simple probability problem, which you should be able to answer using your current knowledge. The stopping rule depends on how much the "Banker" offers the contestant, what are the dollar amounts associated with the remaining unopened cases including the one that the contestant is holding, and the value system of the contestant. Is he/she a risk averse or risk taker? The dilemma here, for each person, is to strike a balance between risk and greed. The same principle, i.e., risk vs. greed, applies to investing in stocks and bonds.
Let's get back to the game. There are 26 cases containing $0.01 , $1 , $5 , $10 , $25 , $50 , $75 , $100 , $200 , $300, $400 , $500, $750 , $1,000, $5,000, $10,000, $25,000, $50,000, $75,000, $100,000, $200,000, $300,000, $400,000, $500,000, $750,000, and $1,000,000.
Let's assume you are the contestant who has opened the following 6 cases in the first round: $100; $1,000; $10,000; $25,000, $300,000, $500,000.
1. What is the expected value (i.e., mean) of the remaining cases?
2. If you are a "risk-neutral" person, how much would the banker has to offer you for you to be indifferent between going to the next round and selling your case for the Banker's offer?
3. What if you are a "risk-averse"?
4. What if you are a "risk-taker"?
Explain your solution in mathematical terms and words. Post a spreadsheet if you wish.
Problem 2:
A multiple-choice test has four possible answers to each of 16 questions. A student, who has not studied for the exam, takes the test and purely guesses the correct answer to each question, (i.e., the probability of getting a correct answer on any given question is 1 out of 4 or 0.25). Answer the following questions in the body of your post and attach your supporting spreadsheet.
a. Can we use binomial distribution here? Are the conditions of the binomial experiment met? Explain why one condition at a time. If we cannot use binomial distribution here, show which condition is violated.
b. If the student purely guesses the answers, what is the expected number of questions he/she would answer correctly?
c. What is the probability that the student does not guess even one question correctly?
d. If at least 11 questions must be answered correctly to pass the test, what is the probability that the student passes the test by purely guessing?
Feel free to attach your spreadsheet or Minitab results. Read the Lecture page and go through its tutorials.