Stat 330
Exam I Practice Questions
1. Short answers
(a) Give an example demonstrating why the number of combinations is always less than the number
of permutations.
(b) List three axioms of probability.
(c) Give two examples which require a continuous sample space.
(d) How does the geometric distribution differ from the Binomial?
2. Counting Examples
(a) A computer shops advertises computers with 5 choices of monitors, three sizes of hard disks and
2 sizes of main memory. At least one computer for each configuration is in the shop. How many
computers have to be in the shop at least? Explain.
(b) In an exam there are 10 questions, each candidate has to choose 3 questions.
How many possibilities to choose three questions are there
i. in total? Explain.
ii. if at least one question has to be chosen from the first 4 questions? Explain.
(c) Someone owns 7 CDs with classical music, 12 CDs with pop music and 5 CDs with Jazz. How
many possibilities are there to put all CDs on a shelf, if all CDs of the same category are next to
each other? Explain.
(d) A group of 11 persons consists of 7 men and 4 women. 5 persons are selected for a committee.
How many possibilities are there
i. in total? Explain.
ii. if exactly one women must be in the committee? Explain.
iii. if at least one women must be in the committee? Explain.
(e) An administrator demands from his users that passwords must be chosen according to the following
rules:
• a password must have exactly 8 characters
• at least one character must be a digit ’0’-’9’ or one of the special characters ’#’,’-’,’ ’,’&’,’%’.
How many possible passwords are there? Explain.
3. Some jobs submitted for processing on a particular CPU have fatal programming errors, while others
do not. Suppose that the long run fraction of jobs with fatal programming errors is p = 0.05.
a) Find (under appropriate distribution assumptions) the probability that among the next 10 jobs
submitted there are less than 3 with fatal errors.
b) One begins monitoring the processing of jobs and lets
X = the total number of jobs processed until the first with a fatal error.
Find (again under appropriate distribution assumptions) P (X ≤ 30)
4. In an experiment you toss two dice. A random variable X is defined as the minimum score of the two
dice.
(a) What is the probability that the minimum of the two dice is 3?
(b) What is the probability that the minimum of the two dice is less than 3?
(c) Give the probability mass function of X. Explain
(d) What is the expected value and the variance of X?
(e) Draw probability mass and probability distribution function of X into the two boxes below. Do
not forget to mark and label the axes appropriately.
5. Given is the following table:
x
pX
−3
0.1
−1
0.15
0
0.3
1
s
(a) Find s, so that pX is a probability mass function.
(b) Assume, the random variable X has probability mass function pX . What is E[X]?
(c) Determine V ar[X].
(d) Assume, Y is another random variable with E[Y ] = 1.25 and V ar[Y ] = 2.5. Evaluate the following
expressions:
i. E[X + 4Y ]
ii. V ar[3Y ]
iii. E[2Y 2 ]
6. In three boxes there are capacitors as shown in the following table:
Capacitance
(in µF )
1.0
0.1
0.01
Number in box
1
2
3
10 90
25
50 30
80
70 90
120
An experiment consists of first randomly selecting a box (assume that each box has the same probability
of selection) and the randomly selecting a capacitor from the chosen box.
a) Draw a tree diagram and determine the probability of selecting a 0.1 µF capacitor.
b) If a 0.1 µF capacitor is chosen, what is the probability that it came from box 1?
Now assume, that the contents of all three boxes are poured into one big box.
c) What is now the probability of selecting a 0.1 µF capacitor from the big box?
d) If three capacitors are selected at random from the big box, what is the probability that at least
one of them is a 0.1 µF capacitor?
7. A manufacturing process produces integrated circuit chips. The fraction of bad chips produced is 20%.
Thoroughly testing a chip to determine whether it is good or bad is expensive, so a cheap test is tried.
All good chips will pass the cheap test, but so will 10% of the bad chips.
(a) Given a chip passes the cheap test, what is the probability that it is a good chip?
(b) If a company using this manufacturing process sells all chips which pass the cheap test, what
percentage of chips sold will be bad?