Exercise 1
Suppose that we have a sample space with five equally likely experimental outcomes : E1,E2,E3,E4,E5.
Let
A = {E1,E2}
B = {E3,E4}
C = {E2,E3,E5}
a. Find P(A), P(B), P(C).
b. Find P(A U B) . Are A and B mutually exclusive?
c. Find Ac, Bc, P(Ac), P(Bc).
d. Find A U Bc and P(A U Bc)
e. Find P(B U C)

Exercise 2
A committee with two members is to be selected from a collection of 30 people, of whom 10 are males and 20 are females.
a. Find the probability that both members are male
b. Find the probability that both members are female
c. Find the probability that one member is male and one is female.

Exercise 3
A warehouse contains 100 tires, of which 5 are defective.
Four tires are chosen at random for a new car.
Find the probability that all four are good.

Exercise 4
In a particular city,
40% of the people subscribe to magazine A, 30% of the people subscribe to magazine B and 50% to magazine C.
However, 10% subscribe to both A and B, 25% subscribe to both A and C, 15% subscribe to both B and C. Finally, 5% subscribe to all three magazines.
A person is chosen at random.
a. What is the probability that the chosen person subscribes to at least one magazine? b. What is the probability that the chosen person subscribes to at least two magazines? c. Find the conditional probability that a person subscribes to magazine A given that he or she subscribes to magazine B.

Exercise 5
Let us consider a student who is taking two tests on a given day. Let A be the event that the student passes the first test and B be the event that he passes the second.
Suppose that :
P(A) = 0.6

P(B) = 0.8

P(A ∩ B) = 0.5

a. Find the probability that the student passes the second test given that he passes the first b. Find the probability that the student passes the first test given that he passes the second

...joint distribution for X and Y .
4.12 If a dealer’s proﬁt, in units of $5000, on a new automobile can be looked upon as a randomvariable
X having the density function
fx= 21-x,0<x<10,elsewhere
ﬁnd the average proﬁt per automobile.
4.14 Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function.
fx= 2(x+2)50<x<10,elsewhere
4.28 Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces
of the product in a cereal box, with
fx= 25,23.75 ≤x ≤26.250,elsewhere.
4.33 Use Deﬁnition 4.3 on page 120 to ﬁnd the variance of the randomvariable X of Exercise 4.7 on page
117.
4.7 By investing in a particular stock, a person can make a proﬁt in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?
4.37 A dealer’s proﬁt, in units of $5000, on a new automobile is a randomvariable X having the density
function given in Exercise 4.12 on page 117. Find the variance of X.
4.12 If a dealer’s proﬁt, in units of $5000, on a new automobile can be looked upon as a randomvariable
X having the density function
fx= 21-x,0<x<10,elsewhere
ﬁnd the average proﬁt per automobile.
4.38 The proportion of people who respond to a certain mail-order solicitation is a random...

...
Event A is rolling a die and getting a 6. Suggest another event (Event B) that would be independent from Event A.
A company runs 3 servers, each providing services to 40 computers. For each server, two of its client computers are infected. What is the probability that 3 randomly chosen client computers serviced by different servers (one per server) will all be infected?
The probability that Alice’s RSA signature on a document is forged is () What is the probability that out of 4 messages sent by Alice to Bob at least one is not forged?
Event A is selecting a “red” card from a standard deck at random. Suggest another event (Event B) that is compatible with Event A.
What is the probability of getting 6 tails in 10 trials of tossing a coin? Solve this problem by using :The approximation mentioned in Theorem 6
The Binomial Distribution
Then compare answers for a) and b) after you have solved the problem.
When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected? Please choose the appropriate method to approximate this quantity.
The probability of an event occurring in each of a series of independent trials is . Find the distribution function of the number of occurrences of in 9 trials. That is, provide a...

....................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
Turn over
2206-7303
0516
–6–
5.
M06/5/MATME/SP1/ENG/TZ2/XX
The probability distribution of the discrete randomvariable X is given by the following table.
x
1
2
3
4
5
P ( X = x)
0.4
p
0.2
0.07
0.02
(a)
Find the value of p.
(b)
Calculate the expected value of X.
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
............................................................................

...complaints can be represented as a table or a graph, both shown below. The randomvariable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.
xi
0
1
2
3
4
5
6
p(xi)
.15
.1
.28
.20
.10
.10
.07
What is the expected number of complaints received per week? Round your answer to two places after the decimal.
Answer
Selected Answer: 2.58
. Question 6 .5 out of 5 points
Deterministic techniques assume that no uncertainty exists in model parameters.
Answer
Selected Answer: True
. Question 7 .5 out of 5 points
Excel can be used to simulate systems that are represented by both discrete or continuous randomvariables.
Answer
Selected Answer: True
. Question 8 .0 out of 5 points
A cumulative probability distribution is often used for simulating the values of a discrete randomvariable.
Answer
Selected Answer: False
. Question 9 .5 out of 5 points
A time series may exhibit a trend and a cyclic behavior at the same time.
Answer
Selected Answer: True
. Question 10 .5 out of 5 points
The exponential smoothing method is particularly used for time series that exhibit only a trend behavior.
Answer
Selected Answer: False
. Question 11 .5 out of 5 points
The number of traffic accidents in a cosmopolitan area during next July is an example of a continuous...

...Week 3: Discrete RandomVariables
Stephen Bush Department of Mathematical Sciences
MM2: Statistics
- Week 3 -
1
RandomVariables
• Reference: Devore § 3.1 – 3.5 • Definitions:
• An experiment is any process of obtaining one outcome where the outcome is uncertain. • A randomvariable is a numerical variable whose value can change from one replicate of the experiment to another.
• Sample means and sample standard deviations are randomvariables
• They are different from sample to sample. • Population means and standard deviations are not random.
MM2: Statistics - Week 3 2
RandomVariables - Examples
• Experiment 1: Pick a student at random from the class
• Let X denote the height of the student
• Experiment 2: Throw a fair dice
• Let X denote the outcome of the dice. X = 1,2,3,4,5, or 6.
• Notice that the outcome of both of these events changes every time you take a new sample.
MM2: Statistics
- Week 3 -
3
1
RandomVariables
• A randomvariable can be continuous or discrete.
• Continuous randomvariables can take any real value, such as measurements. • Electrical current, length, pressure, temperature, time voltage, weight etc. • Discrete random...

...Discrete RandomVariables: Homework
Exercise 1
Complete the PDF and answer the questions.
|X |P(X = x) |X(P(X = x) |
|0 |0.3 | |
|1 |0.2 | |
|2 | | |
|3 |0.4 | |
a. Find the probability that X = 2.
b. Find the expected value.
Exercise 2
Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6.
a. What are you ultimately interested in here (the value of the roll or the money you win)?
b. In words, define the RandomVariable X.
c. List the values that X may take on.
d. Construct a PDF.
e. Over the long run of playing this game, what are your expected average winnings per game?
f. Based on numerical values, should you take the deal?
g. Explain your decision in (f) in complete sentences.
Exercise 3
A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of...

...The number of possible combinations of 3 horses winning, in any order, is
So the probability is
4. In how many distinguishable ways can the letters in the word statistics be written?
5. The table shows the results of a survey that asked 2850 people whether they are involved in any type of charity work.
| Frequently | Occasionally | Not at all | Total |
Male | 221 | 456 | 795 | 1472 |
Female | 207 | 430 | 741 | 1378 |
Total | 428 | 886 | 1536 | 2850 |
A person is selected at random from the sample.
a) What is the probability the person is female or occasionally involved in charity work?
b) Are the events “being female and occasionally involved in charity work” and “being frequently involved in charity work” mutually exclusive?
yes
6. A company gave psychological tests to perspective employees. The randomvariable x represents the possible test scores.
a) Use the histogram to find the probability that a person selected at random from the survey’s sample had a test score of more than two.
b) Find the probability that the person had a test score of at most 2.
7. The following table is a frequency distribution for the number of dogs per household in a small town.
Dogs | 0 | 1 | 2 |
Households | 931 | 297 | 180 |
a) Construct the probability distribution. (round to the thousandths place)
x | 0 | 1 | 2 |
P(x) | 931/1408=0.661 | 297/1408=0.211 |...

...2011
Please use your calculators and give your ﬁnal answers to 3 signiﬁcant ﬁgures. Show your work for full credit. Please state clearly all assumptions made.
1. Classify each randomvariable as discrete or continuous. (a) The number of visitors to the Museum of Science in Boston on a randomly selected day. (b) The camber-angle adjustment necessary for a front-end alignment. (c) The total number of pixels in a photograph produced by a digital camera. (d) The number of days until a rose begins to wilt after it is purchased from a ﬂower shop. (e) The runnning time for the latest James Bond movie. (f) The blood alcohol level of the next person arrested for DUI in a particular county. 2. A bagel shop sells only two diﬀerent types of bagels: plain (P) and cinnamon raisin (C). Five customers are selected at random. Past records have shown that the demand for cinnamon bagels is twice that for plain bagels. Each customer buys only one bagel and the experiment consists of recording what kind of bagel these ﬁve customers buy. Let the randomvariable X be the number of people who buy a plain bagel. (a) Find the probability distribution for X. (b) Suppose at least 3 people buy a plain bagel. What is the probability that exactly 4 people buy a plain bagel? 3. The probability distribution for a discrete randomvariable X is given by the formula p(r) = for r = 1, 2, . . . , 6. (a) Verify...