Download Fall 2009 Exam 2 Review

1. You are required to select a 6-character case-sensitive password for an online account.
Each character may be an upper-case letter, lower-case letter, or a digit from 0 to 9.
How many different passwords can be created under each of the following situations?
(a) There are no restrictions on what each character must be.
(b) The first character may not be a number.
(c) The last four characters must all be different.
(d) There must be at least one capital letter and at least one number.
(e) The password either contains at least one 0 or at least one lower-case vowel Hint: Look at the complement
2. Suppose that two anthropology, four computer science, three statistics, three biology,
and five music books are put on a bookshelf with a random arrangement.
(a) What is the probability that the books of the same subject are together?
(b) What is the probability that all the statistics books are together?
3. Three kids are playing with their Skittles by randomly putting them in a straight line
before eating them. How many distinguishable permutations can each child create?
(a) Cindy has five red, three orange, and two yellow Skittles.
(b) Josh has seven red, three green, and two purple Skittles.
(c) Thomas has four yellow and four orange Skittles.
4. The following questions refer to the letters in the word LOLLIPOP:
(a) How many distinct ways can you rearrange the letters in the word LOLLIPOP?
(b) What is the probability that you get a rearrangement of LOLLIPOP so that the L's
are grouped together?
(c) What is the probability that you get a rearrangement of LOLLIPOP so that the L's
are grouped together and the P's are grouped together?
(d) What is the probability you get a rearrangement of LOLLIPOP so that an L is in
the first spot and a P is in the last spot?
(e) What is the probability that you get a rearrangement of LOLLIPOP so that a
vowel (I or O) is in the 3rd spot?
(f) What is the probability that you get an arrangement of LOLLIPOP so that the I
comes before the two O's?
5. You have 3 quarters, 5 dimes, 1 nickel, and 2 pennies. You randomly grab TWO of the
coins.
(a) What is the probability that you get exactly 30 cents?
(b) What is the probability that you get two coins of the same denomination (i.e.
two quarters)?
6. A group of friends will select two different appetizers from a list of twelve. Four of the
appetizers could be considered 'cold' (like spinach dip) and eight could be considered
'hot' (like mozzarella sticks.) How many different ways could they select two appetizers
under each of the following conditions?
(a) There are no restrictions.
(b) Both appetizers cannot be hot.
(c) There is one hot and one cold appetizer.
(d) Both appetizers must be cold.
7. The discrete random variable X has a PMF described by the table below.
x
pX(x)
2
0.2
4
0.25
6
0.05
8
0.3
10
0.2
(a) What is the probability that X is between 5 and 9?
(b) Given that X is at least 4, what is the probability that X is at least 8?
(c) Calculate the expected value and variance of X.
(d) Let Z = 10X - 5. Find the PMF of Z.
8. Let Y be a discrete random variable with PMF described by the function below
(𝑦+3)2
pY (y) = 174
if y = 2; 3; 4; 5
0 otherwise
(a) Verify that Y has a legitimate PMF.
(b) What is the probability that Y is smaller than 4?
(c) What is the probability that Y is smaller than 4 if we are told Y is not 2?
(d) Find the expected value and variance of Y.
9. The random variable U follows the PMF
pU(u) = k*(5 - u)
if u equals 1, 2, 3, 4 or 5
0 otherwise
(a) Find the value of k
(b) Find the probability that U is any of 2, 3, or 4.
10. A special deck of cards consists of two sets of spades and one set of hearts. A total of 39
cards. The cards are randomly shuffled and the top two are taken off the top of the
deck. The random variable S represents the number of spades among these two cards.
Find the PMF of S.
11. Suppose X and Y are random variables with E(X) = 3, E(Y ) = 4 and Var(X)=2. Find
(a) E(2X + 1)
(b) E(X - Y )
(c) E(X2)
(d) E(X2 - 4)
(e) E((X – 4)2)
(f) Var(2X - 4)
12. Samantha plans to attend a volleyball game and wants to get some of her friends to go
with her. Let X represent the number of the seven friends she calls that are interested.
The probability any one of them will say 'yes' is 0.8, regardless of the responses from the
others. Answer the following questions.
(a) What is the distribution of X and what are its parameter(s)?
(b) What is the probability that at least four of her friends say 'yes'?
13. Identify the parameters p and n for each of the following Binomial distributions and find
the expected value and variance of the random variable described:
(a) The number of heads in 5 tosses of a fair coin.
(b) The number of correct answers on a multiple choice test if each of the 25
questions has 5 possible answers and the student guesses randomly.
(c) The number of 6's in 100 tosses of a fair dice.
14. A pair of fair dice is rolled 10 times. Let X be the number of (double) rolls in which we
see at least one six.
(a) What is the probability distribution of X and what are its parameters?
(b) What is the probability that in exactly five (double) rolls we see at least one six?
15. A store has 50 light bulbs available for sale. Of these, 5 are defective. A customer buys 8
light bulbs randomly from this store.
(a) What is the probability he finds exactly 1 defective bulb among those he
purchased?
(b) What distribution are you using? What are its parameters?
(c) What is the expected number of defective bulbs he will purchase?
(d) What is the variance of the number of defective bulbs he purchased?
(e) What is the probability at most 2 of the bulbs he purchased are defective?
16. A huge warehouse has 5,000,000 light bulbs available for purchase. Of these, 500,000
are defective. A customer buys 8 light bulbs randomly from this warehouse. What is the
approximate probability he finds exactly 1 defective bulb among those he purchased?
17. There are 21 students in a classroom. Fifteen of these students are in the College of
Management, while the other six are not. Suppose we take a random sample of five
students without replacement and let X represent the number of these students who
are in the College of Management.
(a) What is the distribution of X and what are its parameter(s)?
(b) What are the mean and variance of X?
(c) What is the probability that there are more non-Management students in the
sample?
18. A bag contains ten blue marbles and nine red marbles. We reach in and select four at a
time. Find the probability that we get either exactly three blue or exactly three red.
19. NOW, suppose for the previous question we have 10,000 blue marbles and 9,000 red
marbles. Find the approximate probability of getting either exactly three blue or exactly
three red and compare your answer to the previous question.
20. You decide to roll a fair six-sided die until you roll either a 5 or 6 or until you roll 5 times,
whichever comes first. Let X be the number of times you roll.
(a) Construct the PMF for X.
(b) What is the probability you roll between 3 or 4 times inclusive.
(c) Given you have rolled 2 times already and have not yet succeeded, what is the
probability you roll less than 5 times.
(d) Let Z=-5*X+6. Find E(Z) and Var(Z).
21. A barber shop averages 6.5 customers per hour between noon and 5 p.m. Let X
represent the number of customers arriving between 2 and 3:30 p.m.
(a) What is the distribution of X and its parameter(s)?
(b) Find the probability of exactly six customers between 2 and 3:30 p.m.
(c) If you know that no more than six customers arrived between 2 and 3:30 p.m.,
what is the probability there were exactly six?
(d) Suppose the number of customers between 2 and 3:30 p.m. is recorded over a
seven-day period. Assume that the number of arrivals on different days is
independent; give the distribution of Y, the number of days with exactly six
customers between 2 and 3:30 p.m., and its parameter(s).
(e) Find the probability that exactly six customers arrive between 2 and 3:30 p.m. on
at least one of the seven days.
22. To decide how many new cashiers to hire, a large retail store is investigating the flow of
customers to the checkout lanes. Suppose that customers arrive at a checkout register
at an average rate of 7 per hour. Let X be the number of customers arriving at the
checkout lane in a 45 minute period.
(a) Give the distribution of X and its parameter(s).
(b) What is the probability that exactly 5 customers arrive during this period?
(c) What is the probability that there is at least one customer arriving at the
checkout lane during this period?
(d) If more than one customer arrives during this period, what is the probability that
the number of customers is exactly four?