Instructions: You have three hours to complete this exam. You
may use your calculator and normal tables.
1. Suppose the faculty of a small liberal arts college is distributed among ranks and gender as given in the table
Full Professor=F Associate Professor=A Assistant Professor=B
Male=M 0.12 0.45 0.20
Female=W 0.06 0.15 0.02
a) P(F or W)=?
b) P(W)=?
c) P(not an associate professor)=?
d) P(Full professor, if woman)=?
e) P(Female full professor)=?
f) Is probability of being a woman independent of being a full professor? Briefly explain.
2. Suppose X has a binomial distribution with n=23 and p=0.64. Compute each of the following.
a) P(10 < X < 16)
b) P(10 < X < 16)
3. Suppose X has a normal distribution with mean 14 and standard deviation 2.5. Compute each of the following.
a) P(10 < X < 16)
b) P(10 < X < 24)
4. Suppose that a squad of midshipmen consists of 10 third class and 25 others.
a) If eight of these midshipmen are selected at random and without replacement, what is the probability that three or fewer will be third class?
b)If a committee is selected from these midshipmen at random and without replacement, how many should be on the committee for there to be (at least) a 70% chance that the committee contains at least three third class and at least two that are not third class?
5. A test for HIV is given to a population of which 1% have HIV. This test gives a positive indication for HIV 90% of the time, if a person with HIV is tested. It gives a negative indication for HIV 80% of the time, if a person without HIV is tested.
a) What is the probability that the test gives a correct result?
b) If a person test positive for HIV, what is the probability that the person has HIV?
6. Restaurants are inspected by the State of Maryland Health Department (MHD) for a variety of health factors. A restaurant either passes or fails the inspection. Suppose the results of the various restaurants are independent and each restaurant has a 75% chance of passing.
a) If the MHD inspects 30 restaurants in a day, what is the probability 18 or more pass?
b) What is the probability that the first four restaurants will fail?
c) What is the probability that fifteen or more must be inspected to find ten that pass?
7. Suppose an FBI listening post counts the number of telephone calls made by a Russian diplomat. During a typical hour ,the average number of calls is 5.
a) What is the probability that the FBI agent must wait at least 2 hours for 12 phone calls?
b) What is the probability that there will be more than 20 calls in three hours?
c) The FBI agent at the post has an agreement with his relief that they will switch after ten calls. What is the probability that he will work more than three hours before being relieved?
d) At night the rate of calls changes. What is the rate, if there is a 80% probability of waiting at least 3 hours for 2 calls?
8. Suppose the gross income of an Ensign is normally distributed with mean $25,000 and standard deviation $2,000.
a) What is the probability that an ensign's gross income exceeds $30,000?
b) If two ensigns marry, what is the probability that their family income exceeds $60,000?
9. Suppose X is a random variable whose probability mass function is given in the following tables. Suppose Y is independent of X and has the same distribution.
X 1 2 3 P(X=x) 0.40 0.25 0.35
a) What is the probability that X + Y = 4?
b) What is the probability that 2X = 4?
10. Suppose the errors in the measurement of the period of the orbit of Io, a moon of Jupiter, have mean 2.4 and standard deviation 1.7.
a) If 100 independent measurements are made, what is the expected value of the sum of these 100 errors?
b) If 100 independent measurements are made, what is the standard deviation of the sum of these 100 errors?
c) If 100 independent measurements are made, what is the probability that the sum of these errors is greater than 250?
d) What is the probability that the average of these 100 measurements is less than 2.5?
11. Carmen San Diego is hiding in North America with probability 0.4, in Africa with probability 0.35, or in Asia. If Carmen is in North America, she can be found with probability 0.3; if she is in Africa, she can be found with probability 0.5; and if she is in Asia, she can be found with probability 0.6.
a) If all three continents are searched, what is the probability that Carmen is found?
b) If only North America is searched, what is the probability that she is not found?
c) If only North America is searched and Carmen is not found, what is the probability that she is in Asia?
d) If only one continent can be searched, which one should be searched?