1. Consider the Assume that the 26 letters of English language alphabet (A through Z) are the equally likely outcomes (simple events) of a statistical experiment, e.g., a wheel of fortune with 26 equal divisions, each labeled a letter. The 26 different equally probable outcomes comprise the sample space S.
2. following two events, named FIRST and LAST when we conduct the statistical experiment (turning the wheel):
FIRST = (Getting a letter that can be found in your first name)
LAST = (Getting a letter that can be found in your last name)
For example, the person with the name Happy Always would have (no repeated letters):
FIRST = {H, A, P, Y}
LAST = (A, L, W, Y, S)
(a) Define the two events mentioned above more specifically using the actual letters in your first and last names.
(b) Draw a "Venn Diagram" to graphically present your sample space. Your diagram should clearly show whether the two events are sharing any outcomes. The diagram should also show the outcomes that do not belong to any of the two events.
(c) Find P(FIRST)
(d) Find P(LAST)
(e) What is the probability of getting an outcome that belongs to neither of the two events defined?
(f) LAST)ÇDetermine P(FIRST and LAST), which can also be stated as P(FIRST
(g) LAST)ÈDetermine P(FIRST or LAST), which can also be stated as P(FIRST
(h) Are events FIRST and LAST mutually exclusive? Why?
(i) Are events FIRST and LAST independent? Why?
2. A statistical experiment involves flipping a fair coin three times.
(a) Determine the sample space for the statistical experiment. The sample space is the set of all possible outcomes; for example, HTH is one possible outcome, where H stands for Heads and T stands for Tails. The format of your sample space (S) in set notation would look like
S = {HTH, ... }
Hint: You may use a tree diagram to find all possible outcomes.
(b) Construct a two-column probability distribution table reflecting the values of random variable X, representing the number of possible heads (0, 1, 2, or 3), and their corresponding probabilities.
(c) How can you check to verify that you have constructed a correct probability table for the statistical experiment under consideration? Resort to mathematical notations of the governing rules to be brief.
(d) Based on what you have done in part (a) or (b) above, what is the probability of getting heads twice. Explain how the probability was determined.
(e) Find the answer to part (d) above using the binomial probability distribution formula.
(f) How can one find the probability of getting three heads (HHH) without resorting to your sample space, your probability table, or the binomial probability distribution formula? Show your computation and the statistical basis for it.
(g) Using your table, find the expected value of X, as E(X) = ∑(xi P(xi) = x1 P(x1) + x2 P(x2) + ... + xn P(xn).
3. A stamp collector has a set of five different stamps of different values and wants to take a picture of each possible subset of his collection (including the "empty set," depicting just the picture frame!), i.e., pictures showing no stamps, one stamp, two stamps, three stamps, four stamps, or five stamps. In each picture showing two or more stamps, the stamps are in a row. Showing your work, determine the maximum number of different pictures possible, when the difference between two pictures would be either in the number of stamps or in the horizontal order of the stamps (order is important). For example, if the stamp collector had just two different stamps (say A and B) of different values, he would have five pictures showing: A, B, AB, BA, and the empty frame.
4. Twenty six people have been invited to a party. Each invitee meets all other invitees, shaking hands.
(a) Determine the total number of the handshakes.
(b) If four invitees are from out of state and others are local, what is the probability of randomly selecting two invitees and those two are not local.
5. A telemarketing executive has determined that for a particular product, 25% of the people contacted will purchase the product. If 10 people are contacted, what is the probability that at most 2 will buy the product? Show work/explanation.
Hint: The question is not the same as "what is the probability that 2 will buy the product"?
6. The table below gives the distribution of blood types by sex in a group of 1,200 individuals.
Blood Type Male Female Total
O
200 416 616
A
68 284 352
B
40 144 184
AB
12 36 48
Total
320 880 1200
(Answers for parts a through f can be stated as fractions, such as 35/46, or as decimals rounded to three decimal places)
A person is selected at random from the group.
Showing your work, what is the probability that the person:
(a) is female?
(b) has blood type A?
(c) is a female having blood type A?
(d) is a female or has blood type A?
(e) is female, given that the person's blood type is A?
(f) has blood type A, given that the person is female?
Consider the events F = "person is female" and A = "person has blood type A".
(g) Are the events F and A independent? Show work/explain carefully.
7. Using graph(s), shading, and appropriate probability distribution table, and assuming that the heights of women are normally distributed, with a mean of 65.0 inches and a standard deviation of 2.5 inches, and assuming that men have heights that are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches,
(a) Calculate and write down your height in inches.
(b) Determine what percentage of people of your gender would be taller than you are.
(c) If someone of your gender is randomly selected what is the probability that the selected person would be shorter than you are?
(d) What is the minimum height of a person of your gender to join the "Top 15% Tall Club"?
(e) If 100 people of your gender are randomly selected, determine the probability that their mean height would exceed your height.
(f) If 16 people of mixed gender, with a mean height of 67 inches and standard deviation of 2.65 inches are randomly selected from a normally distributed population, construct a 95% confidence interval for the mean of the population.