SM230 Final Examination

 

09 May 97, 0755. Time limit is 3 hrs. You may use calculators and other materials approved by your instructor.

 

1. Calls to 911 operators are initially divided into two categories: police and fire. The calls typically arrive in the following percentages: Fire – 25%, Police - 75%.Thirty percent of all calls will require an ambulance in addition to a police or fire response. Eighty percent of all fire calls will require an ambulance.

a. Fill in the Venn diagram.

Non Ambulance

Ambulance

Fire Calls

Police Calls

b. What fraction of all calls will require both an ambulance and a fire response?

c. What is the probability of needing an ambulance given that the initial call was for police?
  1. Are ambulance calls positively correlated with fire calls and why?
  2. Are ambulance calls positively correlated with police calls and why?

2. A CH-53 crashed while on a routine training mission. The helicopter is on one of three nearby islands. A much more reliable aircraft, the CH-46, is searching for the wreckage. The probabilities of the wreckage being on any one island and the probabilities of finding the wreckage given that it is on a particular island are given below:

P(Island)

P(Detect| Island)

Island A

.2

.7

Island B

.5

.6

Island C

.3

.3

You may search one island a day only.

a. Set up the Venn diagram for the first day of the search. Which island do you search and why?

b. Your first day of searching was unsuccessful. Update the Venn diagram. Which island will you search on the second day and why?

 

3. Recently NMC Bethesda was here with the mobile Blood Bank. If 45% of the general population has Type O blood and 50 people came to donate,

a. What is the probability that the blood bank left with exactly 25 pints of Type O blood?

b. What is the probability that they got more than 25 pints of Type O blood?

c. How many donations would they have needed to be 95% sure of getting at least 25 pints of Type O blood?

 

4. I heard that 20% of the Firsties were able to choose Navy pilot on service selection night. I have a house to rent in Pensacola that will accommodate 3 future pilots.

a. What is the probability that I would have to stop more than 15 Firsties in Bancroft Hall to find 3 who might want to rent my house?

b. How many should I plan to stop to be 90% sure of getting the 3 renters I need?

 

5. Acoustic training tapes are prepared by the intelligence community so that sonar technicians can become familiar with the signals generated by threat submarines. Usually a tape will have either nuclear submarines recorded on it or diesel submarines. A training tape library may have a collection of 24 tapes with nuclear submarine signatures and 18 tapes with diesel submarine signatures. If you randomly select 8 tapes from the library,

a. What is the probability that at least 5 of them have diesel submarine recordings on them?

b. What is the probability that half of them will be nuclear submarine recordings and half will be diesel?

c. What is the probability that you will get at least 3 of each type of tape, diesel and nuclear?

 

6. Enemy subs are in the area but are using a tactic of dispersion that puts on average only 1 sub in 5 cubic miles of water.

a. If I am searching the area for enemy subs by dipping a single sonar device which can see only 1 cubic mile of water, how likely is it that I see a sub?

b. If I search 8 independent areas, what is the chance of no more than 1 detection?

 

 

7. Ships medical receives on average 1 new patient every 7 minutes.

a. What’s the chance medical sees more than 10 patients in the next hour?

b. If medical saw no new patients for an hour, what’s the chance the next patient arrives within the next minute?

 

8. X and Y are independent random variables. Their probability mass functions are given below. W = X + Y.

X

PMF(X)

Y

PMF(Y)

0

0.15

1

0.35

1

0.30

2

0.20

2

0.35

3

0.45

3

0.20

a. What are the possible values for W?

b. What is the probability that W is less than 3?

c. What is the probability that W is 2 or more?

d. What is the mean, or expected value, for W?

e. What is the standard deviation for W?

 

9. From years of intelligence we have learned that enemy ballistic missile submarines transit to their patrol stations along a two leg route. The first leg, from point A to point B, takes 80 hours on average with a standard deviation of 5 hours. The second leg, from point B to C takes 60 hours with a standard deviation of 8 hours. You are the navigator on a SSN assigned to intercept the enemy submarine. You have just learned from the pilot of a CH-46 that an enemy submarine has left point A in transit to its patrol station. Your captain wants to know the following:
  1. At what time would you only have a 50/50 chance of missing the enemy submarine at point B?

b. At what time should you be at point B to ensure that you have a 99% probability of not missing the enemy submarine?

c. What is the mean time and standard deviation of when the enemy submarine will be at point C?

d. At what time should you be at point C to ensure that you have a 99% probability of not missing the enemy submarine?

 

10. At a flight school program, CH-46 pilots are given grades on their skills in three areas: flight operations, safety, and administration. The mean and standard deviation for these grades are as follows: Operations (mean 80, s.d. 5), Safety (mean 90, s.d. 3), Admin (mean 75, s.d. 6). The grades are independent of each other and are summed up to give a total grade.
  1. What is the mean graduating grade?
  2. What is the standard deviation of the graduating grade?

c. If 85 is the minimum safety grade required to graduate, how many pilots do not graduate due to their safety grade?

d. If a student receives less than a 230 combined score, he or she will fail the program and must leave. If seventy students are currently in the program, what is the mean number that will fail?

 

11. Historical statistics indicate that 1 out of 100 sailors tested for drugs test positive. Assuming the historical rates are accurate answer the following if you were in charge of testing an entire aircraft carrier, with CH-46’s embarked of course, of 5000 sailors.
  1. What is the expected number of positive results?
  2. What is the standard deviation?

c. What is the probability of more than 60 sailors testing positive?

d. What is the probability of less than 40 sailors testing positive?

e. What is the probability of 40 to 60 sailors (inclusive) testing positive?

f. What is the maximum number of sailors that test positive 98% of the time?

 

12. Let X be as defined in the table below. Let Z=X2.

 

X

 

1

 

3

 

4

 

6

 

P(X)

 

0.2

 

0.4

 

0.3

 

0.1

a. What is the mean of Z?

b. What is the standard deviation of Z?