G. Fowler, February 16, 1996.

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• This is intended for practice and it is not necessarily similar to the test that you will take. This test should take less than 100 minutes.
• You should work these problems with your book and notes closed. After you have finished it, then seek help from your book, classmates, or instructor.

1. Suppose 30% of Americans have been married at least once, 10% of Americans have been divorced at least once, 50% of Americans are male, 6% of Americans are male and have been divorced at least once, and 13% of Americans are male and have married at least once. If an American is selected at random:
   1. What is the probability that the selected person is male or has been married at least once?
   2. What is the probability that the selected person has not been divorced at least once?
   3. If the selected person has been married at least once, what is the probability that the person has been divorced at least once?
2. There are 26 letters in the alphabet.
   1. How many sequences of 5 letters are there.
   2. How many sequences of 5 letter are there in which no letter is repeated.
   3. How many sequences of 5 letters are there in which one letter is repeated twice and no other letter is repeated?
3. In discussion of how well public schools prepare their students for college, summary "statistics" are quoted. Suppose the school district claims that 80% of its graduates will attend a four year college and 90% of its students graduate. What proportion of its students will attend a four year college?
4. For this problem assume a company contains 32 first class midshipmen of which 20 are male of European decent. Suppose a squad will contain 8 first class midshipmen.
   1. What is the probability that in a randomly squad all the first class midshipmen will be male of European decent?
   2. What is the probability that in a randomly squad exactly 5 of the first class midshipmen be will male of European decent?
5. Suppose a test is given to kindergarten children to determine which level of the first grade they should attend: low, middle or high. Suppose also on the basis of this test 70% of the students are placed in the middle level and 10% are placed in the high level. Of the children in the kindergarten 55% are girls; 8% of the boys are placed in the high level; and 70% of the boys are placed in the middle level.
   1. What proportion of the children placed in the high level are boys?
   2. If there are 200 children in the kindergarten how many boys will be in the high level?
   3. Are the events of being a boy and being placed in the middle level independent?
   4. Are the events of being a boy and being placed in the high level independent?
6. A certain weapon system functions best in calm weather. Assume that it will function satisfactorily with probability 1 at sea state 1.0; with probability 0.8 at sea state 2; with probability 0.6 at sea state 3; with probability 0.4 at sea state 4; with probability 0.0 at sea state 5. The weather prediction for tomorrow is that there is a 10% chance of sea state 1, a 30% chance of sea state 2, a 25% chance of sea state 3, and a 15% chance of sea state 4 and a 20% chance of sea state 5. What is the probability that the system will function satisfactorily tomorrow, if the weather prediction is correct?
7. A high school senior has lost his watch. It is either in his bedroom, his school locker, his car, or some unknown place. If it is in his car, there is a 90% chance that he will find it; if it is in his school locker, there is a 80% he will find it; if it is in his bedroom, there is a 20% he will find it; if it is somewhere else there is a 1% chance that he will find it. There is a 60% chance that it is in his bedroom, a 10% chance that it is in his car and a 25% chance that it is in his school locker. Assume this student's car is at his house and he is at a friend's house.
   1. If he looks in only one place, where should he look and what is the probability that he finds his watch?
   2. He and his friend have dates and not enough time to look both at school and at his house (where both the car and bedroom are). Should they look at school or at his house?
8. Suppose X is a discrete random variable whose possible values are {0,1,2,3,4,,19,20}, i.e., the whole numbers from 0 through 20. If the CDF of X has the values (x/20)² for these values, compute the following.
   1. P(X<10).
   2. P(5<X).
   3. P(4<X<15).
   4. P(X=8).
9. Suppose X is a continuous random variable whose CDF is sin(x) for x between 0 and /2.
   1. Graph the CDF of X for x between -5 and 5.
   2. P(X=0.1)=?
   3. P(0.1<X<0.6)=?
   4. P(0.2<X<2)=?

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