1. At a certain gas station the customers typically buy gas as follows: Regular Unleaded - 50%; Mid Grade Unleaded -15%; Premium Unleaded - 35%. Of these customers, 25% of the regular customers fill their tanks, 60% of the mid grade customers fill their tanks and 80% of the premium customers fill their tanks.

a. Give the Venn diagram for this system.

b. What is the probability that the next customer arriving will fill up?

c. What is the probability that the next customer arriving will fill up with premium?

d. What is the probability that the next fill up will be regular unleaded?

e. Is filling up positively correlated with regular gas?

2. The Navy purchases its 5"-54 rounds from Company A and Company B. The Ammunition comes in three types Armored Piercing (AP), High Explosive (HE) and Antiaircraft (AA). Testing shows the probability of a dud round for each of the companies to be: Company A - AP (0.04), HE (0.01), AA (0.01); Company B – AP (0.02), HE (0.02), AA (0.03). A magazine will typically carry 40% AP, 25% HE and 35% AA. What company will have the lower dud rate if all of the rounds in a magazine are from a single company?

3. The orange army consists of 30% regulars, 40% reservists and 30% draftees. Of the regulars, only 5% are expected to surrender, while 15% of the reservists are expected to surrender and 30% of the draftees are expected to surrender.

a. What fraction of the Army is expected to surrender?

b. Among those who surrender, how many are draftees?

c. Is surrendering independent of being a reservist or draftee? Why?

4. In some ocean areas there may be as many as 6 bottom-mounted hydrophone arrays for detecting submarines. In the "old days" any particular array might have had a 65% chance of holding a submarine contact (being a "hot’ array.) What is the chance that more than 4 of the arrays were "hot" at any particular time?

5. If 15% of the Class of 1999 scored over 700 points on the math SAT and I wanted to talk to a group of 20 of those students,

a. What is the probability that I would have to check the records of more than 150 students to find the 20 that I need?

b. How many records should I plan to check if I wanted to be 99% sure of checking enough to get 20 students?

6. Recently the NMC Bethesda mobile Blood Bank was on board to take blood donations. By the end of the day, they had collected 30 pints of Type O blood and 20 pints of A, B or AB. If the Hospital Corpsman randomly selected 15 pints of blood,

a. What is the probability that she picked out only Type O samples?

b. What is the probability that more than half were Type O ?

c. What is the probability that she picked at least 6 of the Type O and at least 5 of the other types?

7. Enemy reconnaissance aircraft overfly our area on average once every hour.

a. It is 90% likely that we will see at least how many aircraft in a 24 hour period?

b. If we have been watching our radar scopes for 5 hours and not seen an enemy aircraft indication, how likely is it that we get any enemy blip on our scope within the next 2 hours?

8. During the Blob War there is on average one Zeb casualty found every 10 miles of water.

a. If your ship’s morgue has enough room for 8 Zebs, how many miles can your ship sail and still be 90% sure of not exceeding the morgue’s capacity?

b. If your ship’s captain wants to be 90% confident of not exceeding the morgue’s capacity in 100 miles, how many Zebs per mile on average should you tell him will assure this?

9. X and Y are independent binomial random variables. The probability of success for X is 0.55. The probability of success for Y is 0.35. The number of trials for X is 3. The number of trials for Y is 2. W=X+Y.

a. What are the possible values for W?

b. What is the probability that W is less than 4?

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

d. There is a 75% chance that W is less than what value?

e. What is the mean, or expected value, for X?

f. What is the expected value for W?

10. Your are in charge of a maintenance detachment that repairs gas turbine engines on ships upon their arrival into port. Frigates with two engines have a 0.2 probability of repair for each engine. Cruisers with four engines have a 0.15 probability of repair for each engine. Cruisers and destroyers enjoy having CH-46’s deliver ice cream everyday underway. If today you have 2 frigates and one cruiser returning to port, what is the probability that you will:

a. Have no engines to repair?

b. Have exactly 2 engines to repair?

11. Mines are randomly dropped across a channel in two lines, X and Y. The probability that a minesweeper will find a certain number of mines along each of the lines is given by the tables below. The mines were dropped by a CH-46.