Question:

1. How many ways can three items be selected from a group of six items? Use the letters A, B, C, D, E, and F to identify the items, and list each of the different combinations of three items. 

2. How many permutations of two items can be selected from a group of six? Use the letters A, B, C, D, E, and F to identify the items, and list each of the permutations of items A and D, and of C and E. 

3. Suppose N = 20 and S = 10. Compute the hypergeometric probabilities for the following values of n and x. a. n = 4, x = 1 b. n = 4, x = 0 c. n = 4, x = 2 d. n = 4, x = 4 e. n = 4, x = 3 

4. You have eight balls. Four are red and four are green. You pick a ball at random in trial one.

a. Assume you picked a red ball in trial 1. What are your chances of attaining this result?

b. Without replacement (meaning you don''t put any balls back after any trials), what are your chances of getting two red balls in a row (1 in trial 1, 1 in trial 2)?

c. With replacement (meaning you did put back the red ball you picked back), what are your chances of getting two red balls in a row? 

5. Your chances of getting a red ball are 0.25. Your chances of getting a blue ball are 0.40. Your chances of getting a green ball are 0.35. 
a. What are your chances of getting either a red ball or green ball, if you pick a single ball at random? 
b. What are your chances of not getting a red ball or green ball, if you pick a single ball at random? 
c. What are your chances of getting a blue ball, if you pick a single ball at random? 
d. What are your chances of getting a yellow ball, if you pick a single ball at random? 

6. Consider a manufacturing firm that receives shipments of parts from two different suppliers. Let A1 denote the event that a part is from supplier 1 and A2 denote the event that a part is from supplier 2. Currently, 55% of the parts purchased by the company are from supplier 1 and the remaining 45% are from supplier 2. Hence, if a part is selected at random, we would assign the prior probabilities P(A1) = .55 and P(A2) = .45.

The quality of the purchased parts varies with the source of supply. Historical data provides us a defect rate. If we let B denote the event that a part is bad, then we would say that the probability a part from supplier A1 is defective is P(B | A1) = .07. The probability that given a part from supplier A2 is defective is P(B | A2) = .04. Relying on Bayes Theorem, what is the posterior probability that a bad part came from supplier 1? What about from supplier 2? 

7. As an illustration of a discrete random variable and its probability distribution, consider the tardiness of flights at Fort Lauderdale Airport. Over the past 100 days of operation, data show 40 days with no flights late (.4), 10 days with 1 (.1) flight late, 30 days with 2 flights late (.3), 18 days with 3 flights late (.18), and 1 day with 4 flights late (.01), and 1 day with 5 flights late (.01). 
a. What is the mean? 
b. What is the variance? 
c. What is the standard deviation? 

8. Let us consider the lateness of flights at Miami International Airport. On the basis of past experience, the management estimates the probability that any flight will be late is .28. There are only two outcomes, late or not late. The number of trials is equal to four. Calculate the binomial probability distribution for 0 through 4, with 0 meaning that no flights are late, and 4 meaning that 4 flights are late. 

9. Suppose that we are interested in the number of purchases at the drive-through of a Jack In the Box during a 20-minute period on weekday evenings following a Poisson distribution. If we can assume that the probability of a car arriving is the same for any two time periods of equal length and that the arrival or nonarrival of a car in any time period is independent of the arrival or nonarrival in any other time period, the Poisson probability function is applicable. Suppose these assumptions are satisfied and an analysis of historical data shows that the average number of cars arriving in a 20-minute period of time is 9. What is the probability of seven arrivals in 20 minutes? 

10. The flight time of a flight from Washington DC to Miami, Florida can take from 90 minutes to 180 minutes. It follows a uniform probability distribution. 
a. What is the expected flight time or mean flight time -- E(x)? What is Var(x)? 
b. What is the probability a flight will take less than 125 minutes? 
c. What is the probability a flight will take between 135 minutes to 155 minutes? 

11. A man has eight different suits he can wear to work. He also has ten different shirts he can wear with those suits. Under the multiplication rules, how many different arrangements of different shirts to suits can he wear? 

12. You are given the following set of numbers: 
9 9 9 10 10 10 10 11 12 14 17 18 19 21

a. What is the 95th percentile?

b. What is the median?

c. What is the first quartile?

d. What is the third quartile?

e. What is the 78th percentile?

f. What is the 70th percentile?