SM230 Makeup Final Examination
08 May 97, 1930. Time limit is 3 hrs. You may use calculators and other materials approved by your instructor.
1. Customers at a specialized service station initially arrive looking for either brake service or muffler service. Customer service representatives will try to get these people to get an oil change as well. The customers initially arrive in the following percentages: Brakes - 40%; Mufflers - 60%. Thirty percent of all customers will get an oil change as well. Forty percent of all muffler jobs will get an oil change.
a. Fill in the Venn diagram.
|
No Oil Change |
Oil Change |
|
|
Brakes |
||
|
Mufflers |
b. What fraction of all customers get both an oil change and brake service?
c. What is the probability of a customer getting an oil change given that he arrived for a brake job?
d. Are oil changes positively correlated with muffler jobs? Explain.
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 |
.4 |
.7 |
|
Island B |
.4 |
.5 |
|
Island C |
.2 |
.4 |
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. If 20% of the plebes in a Calculus II class of 20 students will be going on a CSTS cruise this summer, instead of YP’s,
a. What is the probability that at least 5 are going sailing?
b. What is the chance that at least 3 but not more than 7 are going sailing?
4. There are 4 ECA’s that need Officer Reps. If there is a 25% chance that an officer will sign up as an O Rep,
a. What is the probability that I will have to ask more than 20 officers to volunteer before I get the 4 that I need?
b. How many officers should I plan to ask if I want to be 95% sure of asking enough?
5. The Naval Academy Duty Officer watchbill is composed of 35 Commanders and Lieutenant Colonels (O-5's) and 25 Lieutenant Commanders and Majors (O-4's). If the Senior Watch Officer randomly picks 30 people to stand duty in June,
a. What is the probability that all the Duty Officers in June will be O-5's?
b. What is the probability that more O-4's are assigned duty in June than O-5's?
c. What is the probability that at least 10 O-4's will have duty in June and at least 10 O-5's will, too?
d. What is the probability that at most 20 O-5 have duty in June?
6. A CH-46, on average, arrives every 5 minutes to pickup cargo pallets while conducting vertical replenishment.
a. If you are in charge of the deck crew and one of your members needs a break for two minutes, what’s the chance the CH-46 returns before your crew member is back?
b. What’s the chance your crew member will miss two CH-46’s during a two minute break?
7. Your ship has numerous redundant components. If the average time to failure of any one of these components is 4 minutes.
a. What’s the chance that a component fails in the next 3 minutes?
b. If 10 similar redundant systems operate independently, what’s the chance that more than 2 systems have a component fail in the next 3 minutes?
8. The probability density function (the derivative of the CDF) for X is:
f(x) = 0.25x + 0.25 0 £ x £ 2
0 otherwise
a. What is the mean, or expected value, for X?
b. What is the standard deviation for X?
c. Find P(X £ 1.5).
9. You are a CH-46 pilot tasked with delivering mail to ships off the coast of Virginia. Your typical route is from Oceana Naval Air Station to a CVN, then to a DDG and back home again. The average time of the first leg from departing Oceana to departing the CVN is 40 minutes with a standard deviation of 5 minutes; the second leg takes 15 minutes with a standard deviation of 3 minutes, and the final leg is 35 minutes with a standard deviation of 4 minutes.
a. What is the mean time and standard deviation for the entire trip?
b. If you depart Oceana at 1300, what is the time interval in which the CVN can be 90% certain of receiving mail?
c. What is the time interval that the DDG can be 95% certain of receiving mail?
d. After you land it takes you exactly 30 minutes to get home with no standard deviation. If you told your spouse that you would be home by 1410, what is the probability that you will be late?
10. The Computer Services division has determined from experience that 20% of plebe issue computers are defective and require replacement. Given the plebe class has 1100 students (who all want to be CH-46 pilots),
a. What is the probability that more than 245 computers will have to be replaced?
b. How many extra computers should Computer Services order to ensure that there is a 95% chance of being able to immediately replace defective computers?
c. If Computer Services only orders an extra 200 computers, what is the probability that they will run out of replacements?
11. Only 2% of 5 inch shell rounds are defective and require a "clearing charge" to remove the shell from the barrel. If a naval ship expected to shoot 1200 rounds of 5" shells at a CH-46 target drone, what is the:
c. probability of needing 20 or fewer clearing charges?
d. probability of needing more than 30 clearing charges?
12. Let X have a Uniform distribution on [-2,2]. Let Y=X2.
a. What is the mean of Y?
b. What is the standard deviation of Y?
c. What is the mean of X+Y?