7.1: Sets

What is a set?

What is the empty set?

When are two sets equal?

What is set builder notation?

What is the universal set?

Example 1: Write the elements belonging to each set.

a. {x|x is a natural number less than 5}

b. {x|x is a state that borders Florida}

What is a subset?

Example 2: Decide whether the following statements are true or false. Explain.

a. {3,4,5,6} = {4,6,3,5}

b. {5,6,9,12} {5,6,7,8,9,10,11}

A property of Subsets:

Example 3: List all possible subsets for each set.

a. {7,8}

b. {a,b,c}

A set with k elements has ________ subsets.

Example 4: Find the number of subsets for each set.

a. {3,4,5,6,7}

b.

What is a Venn diagram? How can we use Venn diagrams to represent sets?

What is the complement of a set?

Example 5: Let U = {1,2,3,4,5,6,7,8,9,10,11} and A = { 1,2,4,5,7} and B={2,4,5,7,9,11}. Find each

set.

a. A’

b. B’

c. ’

d. (A’)’

What is the intersection of two sets?

What is the union of two sets?

What does it mean for sets to be disjoint?

Example 6: Let A = {1,3,5,7,9,11} , B= {3,6,9,12}, C = {1,2,3,4,5} and the universal set

U={1,2,3,4,5,6,7,8,9,10,11,12}. Find each set.

a. A B =

b. A C =

c. (AB) C’ =

Example 7: A department store classifies credit card applicants by gender, marital status, and

employment status. Let the universal set be all credit card applicants, M be the set of male

applicants, S be the set of single applicants and E be the set of employed applicants. Describe

each set in words.

a. M E =

b. M’ S =

c. M’ S’=

7.2: Applications of Venn Diagrams Venn diagrams can be used to represent the intersection and union of sets.

Example 1: Draw Venn diagrams and shade regions to represent each set.

a. A’ B

b. A’ B’

c. A’ (B C’)

Example 2: A researcher is collecting data on 100 households and finds that

76 have a DVD player

21 have a Blu-ray player

12 have both

Answer the following questions.

a. How many do not have a DVD player?

b. How many have neither a DVD player nor a Blu-ray player?

c. How many have a Blu-ray player but not a DVD player?

Example 3: A survey of 77 freshman business students at a large university produced the

following results

25 of the students read Bloomberg Businessweek;

27 read The Wall Street Journal;

27 do not read Fortune;

11 read Bloomberg Businessweek but not The Wall Street Journal;

11 read The Wall Street Journal and Fortune;

13 read Bloomberg Businessweek and Fortune;

9 read all three.

a. How many students read none of the publications?

b. How many read only Fortune?

c. How many read Bloomberg Businessweek and The Wall Street Journal, but not Fortune?

RULE: Union Rule for Sets:

Example 4: A group of 10 students meet to plan a school function. All are majoring in

accounting or economics or both. Five of the students are economics majors and 7are majors

in accounting. How many major in both subjects?

Example 5: The following table gives the number of threatened and endangered animal species

in the world as of May 2014. Source: U.S. Fish and Wildlife Service.

Endangered (E) Threatened (T) Totals

Amphibians and Reptiles (A)

111 56 167

Arachnids and Insects (I)

94 10 104

Birds (B) 291 33 324

Clams, Crustaceans, corals, snails ( C)

134 30 164

Fishes (F) 94 70 164

Mammals 326 40 366

Totals 1050 239 1289

Using the letters given in the table to denote each set, find the number of species in each of the

following sets.

a. E B=

b. E B =

c. (F M) T’

7.3: Introduction to Probability

What is an experiment?

What is a trial?

What is an outcome?

What is a sample space?

Example 1: Give the sample space for each experiment:

a. A spinner like the one shown is spun.

b. For the purpose of a public opinion poll, respondents are

classified as young, middle-aged, or senior, and as male or

female.

c. An experiment consists of studying the numbers of boys and girls in families with

exactly 3 children. Let b represent boy and g represent girl.

What is an event?

Example 2: For the sample space in Example 1(c), write the following events.

a. Event H: the family has exactly 2 girls

b. Event K: the three children are the same sex

c. Event J: the family has 3 girls

An event with only one possible outcome is called a __________________________.

If an event, E, equals the sample space S, then E is called a ________________________.

Example 3: Suppose a coin is flipped until both a head and a tail appear, or until the coin has

been flipped four times, whichever comes first. Write each of the following events in set

notation.

a. The coin is flipped exactly three times.

b. The coin is flipped at least three times.

c. The coin is flipped at least two times.

d. The coin is flipped fewer than two times.

RULE: Set operations for Events:

o Let E and F be events for a sample space S

E F occurs when___________________________________________

E F occurs when __________________________________________

E ‘ occurs when ____________________________________________

Example 4: A study of workers earning the minimum wage grouped such workers into various

categories, which can be interpreted as events when a worker is selected at random. Consider

the following events:

E: worker is under 20;

F: worker is white;

G: worker is female.

Describe the following events in words. Source: Economic Policy Institute.

a. E’

b. F G’

c. E G

DEFINITION: Mutually Exclusive Events

Example of mutually exclusive event:

DEFINITION: Basic Probability Principle:

Example 5: Suppose a single fair die is rolled. Use the sample space S = {1,2,3,4,5,6} and give

the probability of each event.

a. E: the die shows an even number

b. F: the die shows a number less than 10

c. G: the die shows an 8.

For any event E, __________________________________

It is not always possible to establish exact probabilities for events. Instead, useful

approximations are often found by drawing past experience. This is known as

____________________________.

Example 6: The following table lists the estimated number of injuries in the US associated with

recreation equipment. Source: National Safety Council

Equipment Number of Injuries

Bicycles 515,871

Skateboards 143,682

Trampolines 107,345

Playground Climbing Equipment 77,845

Swings or swing sets 59,144

Find the probability that a randomly selected person whose injury is associated with recreation

equipment was hurt on a trampoline.

7.4: Basic Concepts of Probability

RULE: Union Rule for Probability:

Example 1: If a single card is drawn from an ordinary deck of cards, find the probability that it

will be a red or a face card.

Example 2: Suppose two fair dice are rolled. Find each probability.

a. The first die shows a 2 or the sum of the two die is 6 or 7.

b. The sum of the two die is 11 or the second die shows a 5.

RULE: The Complement Rule:

Example 3: If a fair die is rolled, what is the probability that any number but 5 will come up?

Example 4: If two fair die are rolled, find the probability that the sum of the numbers rolled is

greater than 3.

How do we give a probability statement in terms of odds?

Example 5: Suppose the weather forecaster says the probability of rain tomorrow is 1/3. Find

the odds in favor or rain tomorrow.

Example 6: The odds that a particular package will be delivered on time are 25 to 2.

a. Find the probability that the package will be delivered on time.

b. Find the odds against the package being delivered on time.

What is a probability distribution?

Example 8: The following table lists the major holders of US consumer credit (in billions of

dollars) inn 2012. Source: The World Almanac and Book of Facts 2014.

Holder Amount

Depository institutions 1218.6

Finance companies 680.7

Credit unions 243.6

Federal government 526.8

Nonfinancial business 48.5

Pools of securitized assets 49.9

a. Construct a probability distribution for the probability that a dollar of consumer credit is

held by each type of holder.

b. Find the probability that a dollar of consumer credit is held by a finance company or a

credit union.

7.5: Conditional Probability; Independent Events

DEFINITION: Conditional Probability:

Example 1: The chart below shows results of a stock broker survey. Use the information in the

chart to find the following probabilities.

Picked stocks that went up (A)

Didn’t pick stocks that went up (A’)

Totals

Used research (B) 30 15 45

Didn’t use research (B’) 30 25 55

Totals 60 40 100

a. P(B|A)

b. P(A’|B)

c. P(B’|A’)

Example 2: Given P(E)= 0.4 , P(F) = 0.5 and P(EF) = 0.7, find P(E|F)

Example 3: Two fair coins were tossed, and it is known that at least one was a head. Find the

probability that both were heads.

Example 4: Two cards are drawn from a standard deck, one after another without replacement.

Find the probability that the second is a red card, given that the first is a red card.

RULE: Product Rule of Probability:

Example 5: in a class with 2/5 women and 3/5 men, 25% of the women are business majors.

Find the probability that a student chosen from the class at random is a female business major.

Example 6: The Environmental Protection Agency is considering inspecting 6 plant for

environmental compliance: 3 in Chicago, 2 in Los Angeles and 1 in New York. Due to a lack of

inspectors, they decide to inspect 2 plants selected at random, one this month and one next

month, with each plant equally likely to be selected, but no plant selected twice. What is the

probability that one Chicago plant and one Los Angeles plant are selected?

Example 7: Two cards are drawn from a standard deck, one after another without replacement.

a. Find the probability that the first card is a heart and the second card is a red.

b. Find the probability that the second card is a red card.

What is the difference between independent and dependent events?

RULE: If events E and F are independent events, then

RULE: Product Rule for Independent Events:

Example 8: A calculator requires a keystroke assembly and a logic circuit. Assume that 99% of the keystroke assemblies are satisfactory and 97% of the logic circuits are satisfactory. Find the probability that a finished calculator will be satisfactory.

How can we show that two events are independent?

Example 9: On a typical January day in Manhattan, the probability of snow is 0.10, the

probability of a traffic jam is 0.80 and the probability of snow or a traffic jam (or both) is 0.82.

Are the event “it snows” and the event “a traffic jam occurs” independent?

7.6: Bayes’ Theorem

Bayes’ Formula:

Example 1: A manufacturer claims that its test will detect anabolic steroid use (that is, show

positive for an athlete who uses steroids) 95% of the time. What the company does not tell you

is that 6% of all anabolic steroid-free users also test positive (the false positive rate). At ESU, it

is estimated that 10% of all the athletes use anabolic steroids.

Let T denote the event that an athlete tests positive in the drug test and A denote the event

that an athlete has used anabolic steroids. Create a tree diagram to represent the situation.

Then find the following probabilities.

P(A) =

P(T|A)=

P(T|A’)=

P(A’)=

P(T’|A)=

P(T’|A’)=

Compute the probability that an athlete who tested positive is a steroid user.

Example 2: For a fixed length of time, the probability of a worker error on a certain production

line is 0.1, the probability that an accident will occur when there is a worker error is 0.3, and the

probability that an accident will occur when there is no worker error is 0.2. Find the probability

of a worker error if there is an accident.

Example 3: 1 % of women at age forty who participate in a routine screening have breast

cancer. 80% of women with breast cancer will receive a positive mammography. 9.6% of

women without breast cancer will also get a positive mammography.

a. A woman in this age group had a positive mammography in a routine screening. What is

the probability that she has breast cancer?

b. Another woman in this age group has a negative mammography in her routine

screening. What is the probability that she does not have breast cancer?

Example 4: Marie is getting married tomorrow, an at outdoor ceremony in the desert. In recent

years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain

for tomorrow. When it actually rains, the weather man correctly forecasts rain 90% of the

time. When it doesn’t rain, he incorrectly forecasts rain 10% of the time. What is the

probability that it will rain on the day of Marie’s wedding?

Example 5: An insurance company issues life insurance policies in three separate categories:

standard, preferred, and ultra-preferred. Of the company’s policy holders, 50% are standard,

40% are preferred, and 10% are ultra-preferred. Each standard policy holder has probability

0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the

next year and each ultra-preferred policyholder as probability 0.001 of dying in the next year.

What is the probability that a deceased policy holder was ultra-preferred?