You may use calculators and tables.
1. We're going on a YP cruise. 55% of all midshipmen have had a course in advanced CPR. 60% have navigation experience and 17% of mids have both.
(a) What fraction of all mids have neither CPR nor navigation experience?
(b) What fraction of all mids have either CPR or navigation, but not both?
(c) Given that a mid does not have CPR, what is the probability that the mid does have navigation experience?
(d) Are CPR and navigation experience independent? Why or why not?
2. We've gotten underway and can't find the charts we need. They might be on shore (with probability 0.20) or in the nav area (probability 0.50)
or in the captain's quarters (probability 0.30). The probability of detection if we search each area is: 77% for shore, 30% for nav area and 63% for captain's quarters.
(a) Suppose we search all 3 locations. What is the probability of finding the charts?
(b) Suppose we search the captain's quarters only. If we do not find the charts, what is the probability that they are on shore?
3. Suppose that 15% of officers at USNA are YP qualified. We need a total of 6 officers for cruise.
If there are 50 officers in the Division, what is the chance we will find (at least) 6 qualified officers in the Division?
4. Suppose we have located all but 1 officer. We start calling company officers in Bancroft.
(a) What is the expected number of calls we have to make to find the last (qualified) officer?
(b) If we can reach 8 officers in one day, what is the chance we will finish in the first day?
5. YPs break down on average once every 30 days.
(a) We have a 5 day leg ahead of us. What is the chance we can make it without breaking down?
(b) What is the chance we can make a 10 day leg without breaking down?
6. Lifejackets come in Small and Large sizes. We carry 12 large and 7 small. I pick up 5 jackets for a work party.
(a) What is the expected number of Large jackets that I pick up?
(b) What is the probability that I have 3 Large and 2 Small jackets?
7. Weights of midshipmen have a normal distribution with mean 165 lbs and standard deviation of 15 lbs.
(a) What fraction of all mids weighs over 180 lbs?
(b) 75% of all mids have weights in what range? (Give a range in the form 165 +/- x lbs.)
(c) If I put 5 mids in a launch, what is the probability that their total weight is less than 850 lbs?
8. Suppose that the length of bits of rope has an exponential distribution with mean 4 ft.
(a) What is the probability that a bit of rope is at least 6 ft long?
(b) On average, how many pieces of rope will I have to go through before I find one that is at least 6 ft long?
(c) Half of the bits of rope are at least how long?
(d) If I tie 3 bits of rope together, what is the chance that the total length is greater than 10 ft?
(e) Suppose I use one piece of rope to form a circle. Find the cdf for the area of the circle.
9. Suppose we are timing a turn using a digital clock that only shows minutes. After we begin the turn, we continue until the clock changes minutes twice.
(This produces a duration that is between 1 and 2 minutes.) Suppose we begin the turn at a "random" time.
(a) What is the expected duration of a turn?
(b) What is the probability that a turn will last between 1.2 and 1.5 minutes?