12/11/96 - 1330. Time limit is 3 hrs. You may use calculators, normal tables and other materials approved by your instructor.

Shiver me timbers! We be pirates looking for treasure. Arrrrrh!

1. Of all the ships we plunder, 10% carry gold, 15% carry silver and 3% carry both.
   1. How many ships carry neither gold nor silver?
   2. How many carry gold or silver, but not both?
   3. What is the probability a ship carries gold if it is found to carry silver?
   4. If a ship carries gold, what is the chance it carries silver?
2. In our part of the ocean, 2/3 of the ships we encounter are French and 1/3 are English. We know that 10% of French ships carry treasure, while 25% of English ships carry treasure.
   1. What fraction of all ships carry treasure?
   2. If we find a ship with treasure, what is the chance it is English?
3. We've come across another pirate crew's hideout. Their buried treasure might be in the cave, buried on the beach or stashed in the forest. The probability of being in the cave is 55%, being buried on the beach has probability 20% and being in the forest has the remaining probability (25%). If we search the cave, we have a 25% chance of finding the treasure. Searching the beach has probability 10% and searching the forest has probability 70%. (Treat each of the questions below separately. That is, the actions are to be considered individually, not in sequence.)
   1. If we were to search one location first, which should it be and why?
   2. Suppose we decide to start by searching the beach. What is the chance of finding the treasure?
   3. If we search only the beach and do not find the treasure, what is the chance it is in the beach?
   4. Suppose we search the forest and the beach, but not the cave. If we do not find the treasure, what is the chance it is in the cave?
4. Our pirate ship has 21 cannons (on one side). The probability we will hit our target is 0.20 for each of them and they are independent. We will need 5 hits to sink our adversary.
   1. What is the probability of sinking our foe if we fire all 21 cannons at once?
   2. If we fire one cannon at a time until sinking our opponent, how many shots will be required to be 75% sure of sinking him?
5. My crew has 20 men armed with muskets and 25 with sabers. (None with both.) I need at least 3 with muskets and at least 3 with sabers in my attack party.
   1. If I pick 7 men at random, what is the chance of having the required muskets and sabers?
   2. How many should I pick to be 90% sure of having at least 3 of each type? [This is a harder problem.]
6. On our route, we expect to encounter islands every 3.5 days.
   1. What is the chance of encountering 3 or more islands in a week?
   2. What is the chance of encountering no islands in a week?
   3. To the nearest 0.1 days, when should I guarantee my crew of finding an island to be 90% sure of being right?
7. While on our cruise, we meet a whaler. His goal is to catch 4 whales. He gets a whale every 5 days. He brought along enough supplies for a 30 day cruise. Was this wise?
8. Our annoying parrot squawks every 12 minutes.
   1. It's been 5 minutes since the parrot squawked. What is the probability of having 10 more minutes of peace?
   2. We decide to give the parrot 3 more squawks before we ring his bloody neck. What is the chance the parrot will live for 20 more minutes?
9. When the cook dishes out our "stew", we get an amount with a uniform distribution between 1 and 2 cups.
   1. What is the probability of getting more than 1.6 cups?
   2. What is the probability that none of my work crew (8 men) gets more than 1.6 cups?
10. When we find a treasure chest, the weight has a normal distribution with mean 100 lbs and standard deviation 15 lbs.
    1. Our hoist can handle 120 lbs. What fraction of all chests will be too heavy for our hoist?
    2. What capacity hoist would we need to be 99% sure of lifting our chest?
    3. We would like a cart capable of carrying 4 chests at once. What should its capacity be to be 99% sure of being sufficient?
11. Much of the jewelry we "liberate" contains rubies. The probability mass function for the number of rubies per piece of jewelry is given below.
    1. Find the mean number of rubies per piece.
    2. Find the standard deviation for the number of rubies per piece.
    3. Suppose my trove has 100 pieces and my assistant reports the total number of rubies is 101. Should I have him keel-hauled for cheating me?
    4. What is the probability that 2 pieces of jewelry has a total of 5 or more rubies?

# rubies	Probability
0	0.20
1	0.45
2	0.30
3	0.05

Table 1. PMF for rubies