Present value.  Two rival football fans have made the following​ wager: if one​ fan's college football team wins the conference title​ outright, the other fan will donate ​$2000 to the winning school. Both schools have had relatively unsuccessful​ teams, but are improving each season. If the two fans must put up their potential donation today and the discount rate is 7.5​% for the​ funds, what is the required upfront deposit if we expect a team to win the conference title in 5 ​years?  10 ​years?  15 ​years? What is the required upfront deposit if we expect a team to win the conference title in 5 ​years? ​(Round to the nearest​ cent.)

Question 1: Rebel without a cause

Two drivers speed head-on toward each other and a collision is bound to occur unless one of them deviates at the last minute. If both deviate, everything is okay (they both win 1). If one deviates and the other does not, then it is a great success for the driver with iron nerves (he wins 2) and a great disgrace for the deviating driver (he loses 1). If both drivers have iron nerves, disaster strikes (both lose 2).

1. Write down the payoff matrix of this game.

2. Briefly (1-2 sentences) define the notion of Nash Equilibrium.

3. What are the Nash equilibria of this game?

a.) For desktops, the current market share for Windows is 79%. If the operating systems are checked for 20 randomly selected desktops, what is the probability that 17 of them are running Windows? Answer this question without using R or a binomial calculator function. I want you to simplify the combination manually, but a calculator can certainly be used to multiply the combination by ?? ∙(?−?)?−?.
b.) A certain drug treatment cures 90% of cases of hookworm in children. Suppose 50 children suffering from hookworm are to be treated, and all of them receive the treatment. Find the probability that at least 40 are cured.   Answer this question using the pbinom function in R. The command typed into R can either be included as a screenshot or written out by hand.



c.) In California, about 10% of all homeowners are insured against earthquake damage. 30 homeowners are to be selected at random; let X denote the number among the 30 who have earthquake insurance. What is the probability that 3 or less of the 30 selected have earthquake insurance?   Answer this question using the pbinom function in R. The command typed into R can either be included as a screenshot or written out by hand.

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing from year to year, and that the population of the United States is 312 million people.

a) What is the probability that an American chosen at random died as a passenger car occupant last year?

b) What is the probability that you died as a passenger car occupant last year?

  d) What is the probability that an American chosen at random will die as the result of a tornado next year?

g) People sometimes claim skydiving is less dangerous than driving or riding in a car. Does the data support this claim? Explain.

h) People sometimes claim motorcycle riding is less dangerous than traveling by car. Does the data support this claim? What additional information and/or calculations would be useful to evaluate this claim?

Give your answer as a fraction or decimal. If decimal, make sure your answer is accurate to at least 2 significant figures (values after leading zeros)

Federal law under Title 49 of the United States Code, Chapter 301, Motor Vehicle Safety Standard took effect on January 1, 1968 and required all vehicles (except buses) to be fitted with seat belts in all designated seating positions. While most states have laws requiring seat belt use today, some people still do not “buckle up.” Let’s assume that 90 % of drivers do “buckle up.” If drivers are randomly stopped to check seat belt usage, answer the following questions and show your work.

How many drivers do they expect to stop before finding a driver whose seatbelt is not buckled?

What is the probability that the second unbelted driver is in the ninth car stopped?

What is the probability that of the first 10 drivers, 8 or more are wearing their seatbelts?

If they stop 30 cars during the first hour, find the mean and standard deviation of the number of drivers not expected to be wearing seatbelts?

If they stop 120 cars during this safety check, what is the probability they find at least 12 drivers not wearing seatbelts?

(Please type response, handwritten responses can be difficult to understand)

2. Jenni is an insurance agent for the company listed in question 1. The number of whole-life policies she has written (sold) that lapse is Poisson distributed with a mean of 12 policies per year.

a. What is the probability only 9 of Jenni’s whole-life policies will lapse in a given year? (1 point)

b. What is the probability 16 of Jenni’s whole-life policies will lapse in a given year? (1 point)

Question 1 (for reference)

1. An insurance company is concerned about the persistency (length of time a policy remains in-force and doesn’t lapse) of its whole-life policies. Suppose the persistency of the company’s whole-life policies is normally distributed with a mean of 13 years and a standard deviation of 4 years.

a. What is the probability that the persistency of a whole-life policy will be less than 10 years? (1 point)

Answer to A: .2266

b. If the insurance company wants no more than 5% of its whole-life policies to have a persistency of less than 8 years, what does its mean persistency need to be? (Assume the same standard deviation) (1 point)

Answer to B: Mean= 14.5794

IQ Scores are normally distributed with a mean of 100 and a standard deviation of 15. Use this information and a Z-table (or calculator or Excel) to solve the following problems.

A. Ryan has an IQ score of 118. Calculate the z-score for Ryan's IQ.

B. Interpret the z-score you obtained in the previous problem.

C. Suppose an individual is selected at random. What is the probability that their IQ score is less than 111? Round your answer to four decimal places (since this is what is given in the z-table)

D. Suppose an individual is selected at random. What is the probability that their IQ score is greater than 122? Round your answer to four decimal places.

E. Suppose an individual is selected at random. What is the probability that their IQ score is between 95 and 112? Round your answer to four decimal places.

F. MENSA is an organization you can join if you have a high IQ score. MENSA only admits people with IQ scores in the top 2% of people. What is the minimum IQ score you can get and still be admitted to MENSA?  Round your answer to the nearest whole number.

A population of values has a normal distribution with μ=102.8μ=102.8 and σ=41.9σ=41.9. You intend to draw a random sample of size n=88n=88.

Find the probability that a single randomly selected value is greater than 100.6.
P(X > 100.6) =

Find the probability that a sample of size n=88n=88 is randomly selected with a mean greater than 100.6.
P(M > 100.6) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 204.1-cm and a standard deviation of 1.1-cm. For shipment, 27 steel rods are bundled together.

Find the probability that the mean length of a randomly selected bundle of steel rods is greater than 203.6-cm.
P(¯xx¯ > 203.6-cm) =

Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted.