Exercise 1. The mathematics department demands that the chair of the dissertation committee must be a mathematician in order to…
Exercise 1. The mathematics department demands that the chair of the dissertation committee must be a mathematician in order to keep the physicists in check. What? How arrogant! But, there is no other choice if a dissertation committee has to be assembled in time. As usual, physicists have to swallow their pride in order to keep peace. Yes, they are just a bunch of you know what in the mathematics department! For comic relief, can you figure out how many ways can this hapless graduate student can choose among his beloved professors if the chair of the committee must be a mathematician, and the rest of the committee can be a mix of mathematicians and physicists?
Exercise 5. A fair coin is flipped 9 times. What is the probability of getting exactly 6 heads?
Exercise 7. You flip a coin three times.
(a) What is the probability of getting heads on only
one of your flips?
(b) What is the probability of getting heads on at least one flip?
Exercise 9. A jar contains 10 blue marbles, 5 red marbles, 4 green marbles, and 1 yellow
marble. Two marbles are chosen (without replacement).
(a) What is the probability that one will be green and the other red?
(b) What is the probability that one will be blue and the other yellow?
Exercise 27. A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11
plums, and 2 mangos.
(a) Imagine you stick your hand in this refrigerator and pull out a piece of fruit
at random. What is the probability that you will pull out a pear?
(b) Imagine now that you put your hand in the refrigerator and pull out a piece
of fruit. You decide you do not want to eat that fruit so you put it back into the
refrigerator and pull out another piece of fruit. What is the probability that the
first piece of fruit you pull out is a banana and the second piece you pull out is
(c) What is the probability that you stick your hand in the refrigerator one time
and pull out a mango or an orange?
Exercise 86. Roll two fair dice. Each die has six faces.
a. List the sample space.
b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A).
c. Let B be the event that the sum of the two rolls is at most seven. Find P(B).
d. In words, explain what “P(A|B)” represents. Find P(A|B).
e. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification.
f. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification
Exercise 98. At a college, 72% of courses have final exams and 46% of courses require research papers. Suppose that 32% of courses have a research paper and a final exam. Let F be the event that a course has a final exam. Let R be the event that a course requires a research paper.
a. Find the probability that a course has a final exam or a research project.
b. Find the probability that a course has NEITHER of these two requirements
a. Complete the table.
b. What is the probability that a randomly selected child will have wavy hair?
c. What is the probability that a randomly selected child will have either brown or blond hair?
d. What is the probability that a randomly selected child will have wavy brown hair?
e. What is the probability that a randomly selected child will have red hair, given that he or she has straight hair?
f. If B is the event of a child having brown hair, find the probability of the complement of B.
g. In words, what does the complement of B represent?
Exercise 72. You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss.
Exercise 80. Florida State University has 14 statistics classes scheduled for its Summer 2013 term. One class has space available for 30 students, eight classes have space for 60 students, one class has space for 70 students, and four classes have space for 100 students.
a. What is the average class size assuming each class is filled to capacity?
b. Space is available for 980 students. Suppose that each class is filled to capacity and select a statistics student at random. Let the random variable X equal the size of the student’s class. Define the PDF for X.
c. Find the mean of X.
d. Find the standard deviation of X.
Exercise 88. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X. X ~ _____(_____,_____)
d. How many of the 12 students do we expect to attend the festivities?
e. Find the probability that at most four students will attend.
f. Find the probability that more than two students will attend.
Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games.
(For Questions 1, 2, 3, 4 & 5) For 12 pages randomly selected from War and Peace by Leo Tolstoy, the data set lists the mean number of words per sentence, the mean number of characters per word.
1. ( 5 points) Prepare a frequency distribution of words/sentence with a class width of 3 words and another with class width of 4 words.
2. (5 points) construct a histogram of words/sentence with a class width of 3words and another with a class width of 4 words.
3. (5 points) Give a 5-number summary of the words/sentence (minimum, 1-st quartile, 2-nd quartile (median), 3-rd quartile, and maximum), and construct the corresponding boxplot.
4. (3 points) Construct a dotplot of characters/word.
5. (3 points) Find sample mean, median and mode for characters/word.
(For Questions 6, 7, 8, 9 & 10) The National Highway Traffic Safety Administration conducted crash tests of child booster seats for cars. Listed below are results from those tests, with the measurements given in hic (standard head injury condition units). According to the safety requirement, the hic measurement should be less than 1000.
776, 649, 1210, 546, 431, 612
6. (5 points) What is the variance in hic?
7. (4 points) What is the standard deviation in hic?
8. (3 points) Find z-score for hic 1210.
9. (3 points) Find z-score for hic 612.
10. (4 points) Are hic 1210 and 612 unusual values? Justify your answer.