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

a. Compute the probability of receiving four calls in a 5-minute interval of time.

b. Compute the probability of receiving exactly 9 calls in 15 minutes.

c. Suppose, no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time?

d. Suppose, no calls are currently on hold, If the agent takes 5 minutes to complete the current call, what is the probability that no callers will be waiting?

e. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

(1) The table below is a probability distribution of potential quantity of sales of Gourmet sausages during a game. John Bull has to pay a concession fee of $200 to receive a permit to sell sausages at the stadium. Gourmet sausages can be bought at wholesale for $2.00 and sold in the stadium for $3.50 each. Unsold sausages cannot be returned. Given the probability distribution:

1. How many sausages should John Bull expect to sell?
2. How many sausages should John Bull purchase? Gourmet sausages can only be purchased in batches of 50 units as indicated in the probability distribution.

a) Your initial belief about stock A is that its future price cannot be predicted on the basis of existing public information. An insider comes forward claiming that the price will fall. You know the insider is not totally reliable and tells the truth with probability p=0.3. Use Bayes’ theorem to calculate the posterior probability that the stock price will fall, based on the insider’s evidence.A second insider, equally unreliable, comes forward and also claims that the price will fall. Assuming that the insiders are not colluding, what is your posterior probability of a price fall?  Based on your above answers, does the probability of future stock price depend on unreliable insiders? Would you expect this outcome? Explain your argument.

Jobs are sent to a server at a rate of 2 jobs per minute. We will model job arrivals using a (homogenous) Poisson process. For each question, clearly specify the parameter value(s) of the distribution as well as its name. (a) What is the probability of receiving more than 3 jobs in a period of one minute? (b) What is the probability of receiving more than 30 jobs in a period of 10 minutes? (No need to simplify.) (c) What is the expected value and the variance of inter-arrival times? (d) Compute the probability that the next job does not arrive during the next 30 seconds. (e) Compute the probability that the time till the fourth job arrives exceeds 40 seconds.

Suppose the installation time in hours for a software on a laptop has probability density function f(x) = (4/3) (1 − x³ ), 0 ≤ x ≤ 1.

(a) Find the probability that the software takes between 0.3 and 0.5 hours to be installed on your laptop.

(b) Let X₁, . . . , X₃₀ be the installation times of the software on 30 different laptops. Assume the installation times are independent. Find the probability that the average installation time is between 0.3 and 0.5 hours. Cite the theorem you use.

(c) Instead of taking a sample of 30 laptops as in the previous question, you take a sample of 60 laptops. Find the probability that the average installation time is between 0.3 and 0.5 hours. Cite the theorem you use.

A pharmaceutical company has a new drug which relieves headaches. However, there is some indication that the drug may have the side effect of increasing blood pressure. Suppose the drug company conducts a hypothesis test to determine whether the medication raises blood pressure. The hypotheses are:

H₀: The drug does not increase blood pressure.
H_a: The drug increases blood pressure.

Answer the following questions completely:

1. Do you think that for doctors and patients it is more important to have a small α probability or a small β probability? Why?

2. Do you think that the pharmaceutical company would prefer to have a small α probability or a small β probability? Why?

Question 16 options:

Assume that you have placed temperature sensors in different locations in the US. These sensors are set to automatically text you, each day, the low temperature for that day. Unfortunately, you have forgotten whether you placed a specific sensor S in DFW or in Minneapolis (but you are sure you placed it in one of those two places). The probability that you placed sensor S in DFW is 20%. The probability of getting a daily low temperature of 40 degrees or less is 20% in DFW and 80% in Minneapolis. The probability of a daily low for any day is conditionally independent of the daily low for any other day, given the location of the sensor. The sensor stays at a single place throughout your observations, and it cannot change places from day to day (it is stationary).

a) If the first text you got from sensor S indicates a daily low above 40 degrees, what is the probability that the sensor is placed in DFW?

b) If the first text you got from sensor S indicates a daily low above 40 degrees, what is the probability that the second text also indicates a daily low above 40 degrees?

c) What is the probability that the first three texts all indicate daily lows above 40 degrees?

The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $4600. The IRS plans to draw a 1000 sample of tax returns to study the effect of new tax law.

A) What is the probability that the sample mean tax is less than $8100? Round the answer to at least four decimal places. The probability that the sample mean tax is less than $8100 is...?

B) What is the probability that the sample mean tax is between $7500 and $8100? Round the answer to at least four decimal places. The probability that the sample mean tax is between $7500 and $8100 is...?

C) Find the 70^th percentile of the sample mean. Round the answer to at least two decimal places.

D) Would it be unusual if the sample mean were less than $7800? Round answer to at least four decimal places. It would/would not be unusual because the probability of the sample mean being less than $7800 is...?

E) Do you think it would be unusual for an individual to pay a tax of less than $7800? Explain. Assume the variable is normally distributed. Round the answer to at least four decimal places. Yes/No, because the probability that an individual pays a tax less than $7800 is...?

Richard has just been given a 10-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all ten questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?


(b) What is the probability that he will answer all questions incorrectly?


(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.


Then use the fact that P(r ≥ 1) = 1 − P(r = 0).


Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

They should not be equal, but are equal.They should be equal, but may not be due to table error.    They should be equal, but differ substantially.They should be equal, but may differ slightly due to rounding error.

(d) What is the probability that Richard will answer at least half the questions correctly?

. A lie detector will show a positive reading 10% of the time when a person is really telling the truth, and a lie detector will show a positive reading 90% of the time when a person is in fact lying. The lie detector is tested on three subjects. The first subject is (secretly) known to lie to every single question, no matter what the question (Remind you of anyone?). The second lies to 70% of all questions, and the third lies to 10% of all questions.

a. What is the probability that the lie detector shows a positive on all three subjects?

b. What is the probability that the lie detector shows a negative on all three subjects?

c. What is the probability the lie detector shows a positive on the first subject and a negative on the other two subjects?

d. Given that the lie detector showed a positive result on exactly one subject, what is the probability that subject was the first one?

e. Given that the lie detector showed a positive result on exactly two subjects, what is the probability those two subjects were subjects 1 and 2?

f. Given that the third subject told the truth, what is the probability the lie detector incorrectly concludes he lied? This is called a false positive reading.