University of North Carolina, Greensboro

ECO 25

CHAPTER 3

1)The following table shows a partial probability distribution for the MRA Company’s projected profits (in thousands of dollars) for the first year of operation (the negative value denotes a loss):

1. Find the missing value of f (200). What is your interpretation of this value?
2. What is the probability that MRA will be profitable?
3. What is the probability that MRA will make at least $100,000?

2. Data were collected on the number of operating rooms in use at Tampa General Hospital over a 20-day period. On 3 of the days only one operating room was used; on 5 days, two were used; on 8 days, three were used; and on 4 days all four rooms were used.

1. Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day.
2. Draw a graph of the probability distribution.
3. Show that your probability distribution satisfies the requirements for a valid discrete probability distribution.

3. Brandon Lang is a creative entrepreneur who has developed a novelty soap item called Jackpot to target consumers with a gambling habit. Inside each bar of Jackpot shower soap is a single rolled-up bill of U.S. currency. The currency (rolled up and sealed in shrink-wrap) is appropriately inserted into the soap mixture prior to the cutting and stamping procedure. The distribution of paper currency (per 1000 bars of soap) is given in the following table.

          Distribution of Paper Currency Prizes                                

1. What is the expected amount of money in single bar of Jackpot soap?
2. What is the standard deviation of the money in single bar of Jackpot soap?
3. How many bars of soap would a customer have to buy so that, on average, he or she has purchased three bars containing a $50 or $100 bill?
4. If a customer buys 8 bars of soap, what is the probability that at least one of these bars contains a bill of $20 or larger?

4. The demand for Carolina Industries’ product varies greatly from month to month. Based on the past two years of data, the following probability distribution shows the company’s monthly demand:

1. If the company places monthly orders equal to the expected value of the monthly demand, what should Carolina’s monthly order quantity be for this product?
2. Assume that each unit demanded generates $70 in revenue and that each unit ordered costs

$50. How much will the company gain or lose in a month if it places an order based on your answer to part (a) and the actual demand for the item is 300 units?

3. What are the variance and standard deviation for the number of units demanded?

5. The J. R. Ryland Computer Company is considering a plant expansion that will enable the company to begin production of a new computer product. The company’s president must determine whether to make the expansion a medium- or large-scale project. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.50, and 0.30, respectively. Letting x indicate the annual profit in $1000s, the firm’s planners developed profit forecasts for the medium- and large-scale expansion projects.

Medium scale ExpansioLnarge scale Expansion

1. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit?
2. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?

6. A survey on British Social Attitudes asked respondents if they had ever boycotted goods for ethical reasons (Statesman, January 28, 2008). The survey found that 23% of the respondents have boycotted goods for ethical reasons.

1. In a sample of six British citizens, what is the probability that two have ever boycotted goods for ethical reasons?
2. In a sample of six British citizens, what is the probability that at least two respondents have boycotted goods for ethical reasons?
3. In a sample of ten British citizens, what is the probability that none have boycotted goods for ethical reasons?

7. When a new machine is functioning properly, only 3% of the items produced are defective. Assume that we will randomly select two parts produced on the machine and that we are interested in the number of defective parts found.

1. Describe the conditions under which this situation would be a binomial experiment?

2. How many experimental outcomes yield one defect?
3. Compute the probabilities associated with finding no defects, one defect, and two defects?

8. Military radar and missile detection systems are designed to warn a country of enemy attacks. A reliability question deals with the ability of the detection system to identify an attack and issue the warning. Assume that a particular detection system has a 0.90 probability of detecting a missile attack. Answer the following questions using the binomial probability distribution:

5. What is the probability that one detection system will detect an attack?
6. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
7. If three systems are installed, what is the probability that at least one of the systems                                                                                                                                                                   will detect the attack?
8. Would you recommend that multiple detection systems be operated? Explain.

9. Consider a Poisson probability distribution with 2 as the average number of occurrences per time period…

1. Write the appropriate Poisson probability function?
2. What is the average number of occurrences in three time periods?
3. Write the appropriate Poisson probability function to determine the probability of x occurrences in three time periods?
4. Find the probability of two occurrences in one time period?
5. Find the probability of six occurrences in three time periods?
6. Find the probability of five occurrences in two time periods?

11. Telephone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.

1. Find the probability of receiving 3 calls in a 5-minute interval?
2. Find the probability of receiving 10 calls in 15 minutes?
3. Suppose that no calls are currently on hold. If the agent takes 5 minutes to complete processing the current call, how many callers do you expect to be waiting by that time?
4. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted?