Chapter 7: Sets and Probability Part 1: Sets · Chapter 7: Sets and Probability Part 1: Sets 7.1 Sets What is a set? A set is a _____ collection of objects. We should always be able

Finite Math B: Chapter 7 Notes Sets and Probability 1

Chapter 7: Sets and Probability Part 1: Sets

7.1 Sets What is a set? A set is a __________________________ collection of objects. We should always be able to answer the question: “Is object X in this set or not?” Examples of “sets” {Mrs. Leahy’s Semester 2 Discrete Math Class} {Coins minted by the US Treasury in 2011} {even integers} Generally in this chapter we will be talking about sets of numbers. Set Notation: Use set braces: { } to enclose the numbers in set.

Naming Sets: Sets are often named with letters: 10,20,30A

An empty set is a set with __________________________. Symbol: _____ Can you think of a “collection” or group that would have no members? The universal set is a set that contains all the objects being discussed. (integers, people, whole numbers, etc.)

Example 1: 1,3,5,7,9

The numbers 1, 3, 5, 7, 9 are called the __________________ or _________________ of the set.

Use the symbol: 3 1,3,5,7,9 SAY: “3 is an element of the set”

Use the symbol: 4 1,3,5,7,9 SAY: “4 is NOT an element of the set”

Example 2: 10,20,30A

True or False? 11 A 25 A 10 A


CAUTION: 0 = the number zero

= an empty set, a set with no elements

0 = a set that has one element – the number zero

= a set that has one element – the empty set

Two sets are equal if they contain ______________ the ____________ elements.

NOTE: If Set A equals Set B, we say

A B If Set A does not equal Set B, we say:

A B

Set-Builder Notation: Useful when we are looking for objects that share a common property

has property Px x

SAY: “The set of all elements x such that x has property P”

Example 3: How many elements are in each set?

Symbol: ( )n A = the number of elements in a finite set A.

a. 5,6,7A b. B {positive integers less than 5} c. 0C

Example 4: True or False?

a. {1,10,100} {100,1,10}

b. {32,33,34} {32,33,34,35}

c. positive even numbers 8 {0,2,4,6,8}

d. 3,5,7,9 5,7,9,3

Example 5: is a whole number between 7 and 10x x

Elements: _________________________

Finite Math B: Chapter 7 Notes Sets and Probability 3 *****

Subsets

Consider the two sets A and B: {3,4,5,6}A {2,3,4,5,6,7,8}B

Notice that EVERY element is A is also an element in B. We say that A is a subset of B.

The empty set is a subset of every set. A set is always a subset of itself.

Example 6: {1,2,3,4}

{1,2,3,4,5}

A

B

True or False:

A B

A B

B A

B A

Example 7: {5,6,7,8}

{7,6,5,8}

A

B

True or False:

A B

A B

B A

B A

Example 8: {1,2,3,4,7}

{1,2,3,4,5,12}

A

B

True or False:

A B

A B

B A


Example 10: Find the number of subsets for each set.

a. {9,8,7,6} b. is a day of the weekx x c.

Counting Subsets

Basically, there are two different possibilities for each element = yes or no SO:

Example 9: How many subsets are possible? What are they? (Tree diagrams can sometimes be useful)

a. 5,6

b. , ,x y z


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

Find each set.

a. 'A c. '

b. 'B d. ' 'A

Venn Diagrams: Helps to show elements of sets & subsets. U is the universal set A is a subset of B. B is a subset of U.

Set Operators: Often we will form new sets by combining or manipulating one or more existing sets. Complement: The elements in the universal set NOT in your set. A (white) is a set. A’ (pink) --- say “A prime” is everything else and is called the complement of A. For Example: Let U = the students in a class A = the set of all female students in the class. A’ = the set of all male students in the class.

Finite Math B: Chapter 7 Notes Sets and Probability 6 Intersection: The intersection of two sets is the set of all elements belonging to BOTH A and B.

Symbol: A B Example: Let U = the students in a class A = the set of all male students B = the set of all students with “B” averages.

Then A B = the set of all male students with “B” averages.

Disjoint Sets: Disjoint sets have no elements in common. The intersection of disjoint sets is the empty set.

Example: {1,2,3,4} {5,6,7,8}

Example 11: Let 1,2,3A , {1,3,5,7}B , and the universal set {1,2,3,4,5,6,7,8}U

Find each set:

a. A B b. 'A B

Finite Math B: Chapter 7 Notes Sets and Probability 7 Union: The set of all elements belonging to set A, to set B, or to both sets.

Symbol: A B Example: Let U = students in a class A = set of all female students B = set of all students with B averages

A B = any student who is either female OR has a “B” average

Example 12: Let {1,2,3,4,5}A , {1,3,5,7,9}B , and {1,2,5,6,9,10}C

for the universal set { is an integer such that 1 12}U x x x

Find each set:

a. A B b. 'A C

c. A B C d. ( ' ) 'A B C

Finite Math B: Chapter 7 Notes Sets and Probability 8 *****

Applications of Sets Example: Pg 348 The following table give the 52-week high and low prices, the closing price, and the change from the previous day for six stocks in the Standard & Poor’s 100 on April 11, 2006. Let the universal set U consist of the six stocks listed in the table. Let A contain all stocks with a high price greater than $34. Let B contain all stocks with a closing price between $26 and $30. Let C contain all stocks with a positive price change.

State the elements in each set: A = A’ =

B = A C =

C = A B = Example: A department store classifies credit applicants by gender, marital status, and employment status. Let the universal set be the set of all applicants, M be the set of all 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

d. 'M E


7.2 Applications of Venn Diagrams pg 353 “The responses to a survey of 100 household show that 21 have a DVD player, 56 have a videocassette recorder, and 12 have both. How many have neither a DVD player nor a videocassette recorder?” Venn Diagrams are very useful for “sorting out” this information. Things that might happen:

Example 1: Shade the following Venn Diagrams

a. A B b. A B c. ' 'A B

1 set leads to 2 regions 2 sets lead to 4 regions 2 sets lead to 3 regions

2 sets lead to 3 regions 3 sets lead to 8 regions

A

BC

A

BC

A

BC


d. ' 'A C e. ' ( ')A B C f. B A C

Example 2a: “The responses to a survey of 100 household show that 21 have a DVD player, 56 have a videocassette recorder, and 12 have both. How many have neither a DVD player nor a videocassette recorder?”

Example 2b: Mrs. Leahy surveys the 35 students in her period 2 finite math class. She notes that 15 students said they like donuts, 25 students say they like bagels, and 5 students were absent the day of the survey were able to respond. Assuming that everyone who was present answered, how many students like BOTH bagels and donuts?

A

BC

A

BC

A

BC

In general: These questions are often referred to as Both/Neither Questions and usually have information about two types of elements: Type A and Type B. Total = Type A + Type B + Neither – Both

Finite Math B: Chapter 7 Notes Sets and Probability 11 Example 3: A survey of 77 freshman business students at a large university produced the following results. 25 read Business Week 19 read The Wall Street Journal 27 do not read Fortune 11 read Business Week but not The Wall Street Journal 11 read The Wall Street Journal and Fortune 13 read Business Week and Fortune 9 read all three The Union Rule For Sets:

Essentially:

n(A) + n(B) x y z y

In general, if you feel like you are “missing” information in a survey problem, you need probably need to use the union rule in some way. Example 4: A group of 10 students are all majoring in either accounting or economics or both. Five of the students are economics majors and 7 are majors in accounting. How many major in both subjects?

Questions to answer: How many read none of the publications? How many read only Fortune? How many read Business Week and The Wall Street Journal but not Fortune?

Finite Math B: Chapter 7 Notes Sets and Probability 12 Example 5: The table gives the number of threatened and endangered animal species in the world as of April 2006. (pg 358) 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 c. ( ) 'F M T

b. E B Example 6: Suppose that a group of 150 Students have joined at least one of three chat rooms: one on auto-racing, one on bicycling, and one for college students. For simplicity, we will call these rooms A, B, and C. In addition, 90 students joined room A 50 students joined room B 70 students joined room C 15 students joined rooms A and C 12 students joined rooms B and C 10 students joined all three rooms How many students joined both A and B?

Finite Math B: Chapter 7 Notes Sets and Probability 13 Example 7: A survey was conducted of 150 High School students who were asked about which international city they would like to visit from Athens, Dublin, and Hong Kong. The results were tallied as follows: 60 students wanted to visit Athens 70 students wanted to visit Dublin 80 students wanted to visit Hong Kong 10 students did not respond to the survey 25 wanted to visit both Athens and Dublin 15 wanted to visit both Athens and Hong Kong 35 wanted to visit both Dublin and Hong Kong How many students responded that they wanted to visit all three cities?


7.3 Intro to Probability Vocabulary:

Experiment: an activity or occurrence with an observable result

Trial: Each repetition of the experiment

Outcome: A possible result of a trial

Sample Space: The set of all possible outcomes of that experiment Event: a subset of the sample space Example: Experiment – Tossing a coin Sample space: Example: Give the sample space for each experiment a) Rolling a die b) Drawing a card from a deck containing only the 13 spades c) Measuring the weight of a person to the nearest half-pound (the scale will not measure more than 300 lb) d) Tossing a coin 3 times


Example: For each sample space, write the set the represents the given event a) Sample Space: Rolling a die Event 1: The die is even

Event 2: The die shows a multiple of 3 Event 3: The die shows a 1 Event 4: The die shows a number less than 4 b) Sample space: A family with three children Event 1: The family has exactly two girls Event 2: The family has three children of the same sex Event 3: The family has at least one boy

Set Operators: Remember

E AND F both occur: E OR F or both occur: E does not occur: E and F are DISJOINT (Mutually Exclusive):


Probability Probability is the likelihood of an event: 0 1 Let S be a sample space of equally likely outcomes and let event E be a subset of S, Then the probability that the event E occurs is:

( )

( )( )

n EP E

n S

Example: Find the Following Probabilities 1. You roll two fair dice a. P(a sum of 5) b. P(a sum less than 5 OR a sum greater than 10) c. P (a sum greater than or equal to 10 AND a sum that is even) 2. You draw a card from a standard deck of 52 playing cards a. P (a heart) b) A red queen c) a face card d) P (a heart OR a queen) e) P(a heart AND a queen)


7.4 Basic Concepts of Probability Reminder: intersection: A B union: A B

When dealing with probability you often have to be very careful about overlap. We will use the union rule to help us.

Basically: The probability of E or F happening is equal to the probability of E + the probability of F – the probability of the overlap of the two events. Example 1: You have a box with several geometric shaped tiles that are lettered A, B, or C. (See picture above) You draw out a single tile from the box. Find the following probabilities. a) P(triangle or square) b) P(triangle or letter B) c) P(square or letter C) d) P(square or letter A) e) P(circle and letter C) f) P(triangle and letter C) Example 2: You draw a card from a standard deck of 52 playing cards. Let R represent the event “draw a red card” , let F represent the event “draw a face card”, and let J represent the event “draw a Jack.” Find the following probabilities.

a) ( )P R J b) (J F)P c) (F R)P

A

A

A

A

A

B B

B

B

C

C

C

Union Rule for Probability For any events E and F from a sample space S:

( ) ( ) ( ) ( )P E F P E P F P E F


Complements Often in probability it is easier to find the complement of an event than the event itself.

Example: Two fair dice are rolled and their sum is calculated. Find the following probabilities.

a) P(sum is not 12) b) P(sum is larger than 3)

Odds vs. Probability Sometimes probability is given in terms of odds.

Example: You have a 2 out of 5 chance that you will win a door prize. What are the odds in favor of you winning the prize?

You can also turns odds back into probability by determining ( )n S

Example: You read that the odds of contracting the flu this year are 7 to 50. What is the probability that you will get the flu?

Example: The following table lists the probability that a dollar spent by US Advertisers is spent on a particular medium. a) P(Newspapers) b) P(Newspapers or Broadcast TV) c) P(Not Magazines nor Yellow Pages)

Direct Mail 0.1979

Newspapers 0.1767

Broadcast TV 0.1754

Cable TV 0.0816

Radio 0.0742

Yellow Pages 0.0531

Magazines 0.0464

Other 0.1946

Complement Rule for Probabilities

( ) 1 ( )P E P E

Odds in favor of an Event E

The ratio of ( ) : ( )P E P E

( )

( ')

P E

P E where ( ') 0P E

If the odds in favor of an event E are m to n, then

( )m

P Em n

and ( ')n

P Em n


Properties of Probability You are told the responses to a survey were as follows: 30% yes 40% no 35% undecided Problem? The probability of any event must be between _____ and ______ . No exceptions. The SUM of all the probabilities of events in a sample space must equal ______ or _______. (rounding could cause issues with this in the tiny decimals, so be careful) Example: Are these possible outcomes of an experiment? Why or why not? a)

Outcome S1 S2 S3 S4 S5

probability .3 .3 .3 .1 .1

b)


probability 1.5 -0.5 -0.5 0.3 0.2

c)


probability .3 .1 .1 .4 .1


7.5 Conditional Probability A manager at a brokerage firm notices that some of his stockbrokers follow the company’s suggestions based on research while others tend to operate on more of a “gut feeling” and make their own calls. Using the data in the chart, find the following probabilities. 1. P(a stockbroker picked stocks that went up) 2. P(a stockbroker did not pick stocks that went up) 3. P(a stockbroker used research) 4. P(a stockbroker didn’t use research) What if the manager wants to compare the success of his brokers who are using research to those that are not? 5. P(a stockbroker who used research 6. P (a stockbroker who didn’t use research picked stocks that went up) picked stocks that went up)

When you reduce your _____________________ you are using a concept called “Conditional Probability.”

So for our example: 5. P(Stocks went up|used research)= P(A|B) 6. P(Stocks went up|did not use research)= P(A|B′)


Example 1: Given the following Venn Diagram a) 𝑃(𝐸) = b) 𝑃(𝐹) = c) 𝑃(𝐸 ∩ 𝐹) = d) 𝑃(𝐸|𝐹) = e) 𝑃(𝐹|𝐸) = Example 2: Suppose you ask 100 people their age and what they prefer to drink at lunch and record the results in the table to the right. Find the following probabilities. a) P(soda) b) P(40-49) c) P(30 or older) d) P(soda or coffee) e) P(coffee or 20-29) f) P(coffee and 20-29) g) P(𝑠𝑜𝑑𝑎|30 − 39) h) P(50 − 59|𝑡𝑒𝑎) i) 𝑃(𝑤𝑎𝑡𝑒𝑟|𝑢𝑛𝑑𝑒𝑟 40)

Finite Math B: Chapter 7 Notes Sets and Probability 22 Example 3: A single card is drawn from a deck of 52. a) P(A Queen given the card is black) b) P(a face card given the card is a heart) Example 4: A fair die is rolled. a) P( a 2, given the number is even) b) P(a 1, given the number was less than 4)

Product Rule of Probability:

Example 3: You know that 54% of your high school is female and 46% of your high school is male. A survey in the school newspaper states that 32% of female students are planning to go to prom. 45% of the male students are planning on attending prom. If you randomly choose a name from the school roster, what is the probability…. P(female attending prom) = P(male not attending prom) = Example 4: You draw one card from a standard deck of cards. Use the Product rule to find the probability that the card is a red queen.


Conditional probability may need to be considered with independent/dependent events. For example: You draw a card from a deck and, without replacing it, you draw a second card. 1st draw: 2nd draw: You flip a coin. You flip the coin again. 1st flip: 2nd flip: Events are considered to be independent if:

Are the following events independent? Drawing a card from a deck, without replacing it, drawing another card. Drawing a card from a deck, replacing it, drawing another card. Flipping a coin twice. Flipping a coin. Rolling a die.

Choosing a name off the course roster to be the president. Choosing another name to be the vice president.

Finite Math B: Chapter 7 Notes Sets and Probability 24 Example 5: You draw a card from a standard deck and then without replacing it, you draw again. Find the following probabilities. a) the second card is red given the first card was red b) the second card is red given the first card was black c) the second card is an ace given the first card is a queen d) the second card is a heart given the first card is a heart Example 6: You draw a card from a standard deck and then REPLACE IT before drawing a second card. Find the probability. a) the second card is an ace given the first card is a queen b) the second card is a heart given the first card is a heart Example 7: You draw a card from a standard deck and without replacing it, you draw a second card. Find the probability. a) Both cards are hearts. b) The first card is a king and the second card is a queen.

Documents

Chapter 7: Sets and Probability Part 1: Sets · Chapter 7: Sets and Probability Part 1: Sets 7.1 Sets What is a set? A set is a _____ collection of objects. We should always be able