You are the plane commander for a P-3 that is participating in an ASW exercise off the east coast.

1. Your crew is loading the sonobuoys that will be used during your mission. The sonobuoys are stored on racks and they are all mixed up. All that the crew members know is that the racks hold 40 sonobuoys for passive detection and 10 sonobuoys for active detection.

a. If your crew loads 20 sonobuoys from the racks, what is the chance that they get the 16 passive sonobuoys that you need to lay a proper search pattern?

b. How many should they load to be 95% sure of having the 16 you will need?

2. Let's say that we load 30 sonobuoys from the rack. We'll check them during the time it takes to get on station to determine which are passive and which are active sonobuoys. If a sonobuoy has an 80% chance of being passive,

a. What is the probability that we will have to check more than 20 of the unidentified sonobuoys that were loaded to be able to identify 16 passive sonobuoys?

b. Once we have checked all 30 of the sonobuoys that were loaded, what is the probability that we will have at least the 20 passive sonobuoys that we need and at least 6 active sonobuoys (just in case we want to "ping" on a contact)?

3. Passive sonobuoys that are used in a general search last (on the average) 8 hours. We are scheduled to lay a search pattern and then remain on-station (on-sta) to monitor it for 8 hours. To maintain "pattern integrity", sonobuoys that fail are replaced.

a. What is the probability that a single sonobuoy will last for an entire on-sta period?

b. With 90% certainty, what is the maximum number of times you should expect to replace the sonobuoy in any particular location while you are on-station?

c. If we have been on station for 2 hours without any sonobuoy failures, what is the probability that we will not have a failure for another 3 hours?

4. Sometimes one P-3 will complete their 8-hour on-sta period and be relieved by another that will continue to monitor the search pattern that was already there. What is the probability that the sonobuoy in one specific location will be replaced for a 3rd time within 12 hours of original placement?

5. Assume that when a sonobuoy fails, it is not replaced. What is the probability that there will be at least 10 functioning sonobuoys at the end of the on-sta period?

6. Match the description with its appropriate distribution(s).

I. Binomial

II. Negative Binomial

III. Hypergeometric

IV. Poisson

V. Exponential

VI. Erlang

A. P = prob(success) is one of the parameters

B. Has discrete random variables

C. The random variable is time to complete R events (where R>1)

D. Is memoryless

E. Probability problems are solved by conversion to a Poisson distribution

F. R = # of successes is one of the parameters

G. X = # of desired elements in the subset selected