cferreras.weebly.com · Web viewThe ___ (set with no elements) is a subset of every set. The _ (or ___) is denoted by the symbol { } or the symbol ∅ . In addition, any set

$Page 1: cferreras.weebly.com · Web viewThe _____ (set with no elements) is a subset of every set. The _____ (or _____) is denoted by the symbol { } or the symbol ∅ . In addition, any set$
Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolDay 1: An Introduction to Sets & Probability

Warm-Up Cassettes CDs Total

ClassicalPop

Total

In Mr. Stamp’s music collection there are: 84 cassettes and CDs in total 58 popular cassettes and CDs 2 classical cassettes out of 50

Complete the above table. What percentage of his collection are classical CDs?

Complete the table for the activities chosen by 74 teenagers on an activity holiday Rock

ClimbingMountain Climbing

Total

Boys 5Girls 7 20Total

What percentage of people were boys who went rock climbing?

A ______ is a collection of objects called __________. The elements (or members) of a set are often numbers, letters, geometric shapes, or names. You have been exposed to the term set in this course already (for example—the solution set of an equation)

If a set consists of a “small” finite list of elements, then we donate the set in _______ (list) form by listing its elements inside set braces, {…}, separated by commas.

Example 1: In roster form, write sets that match the following descriptions:

(a) all even integers greater than zero and less than 10

(b) all months whose names begin with the letter J

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School

In order to refer to specific sets, we typically name them with capital letters. For example, we could define sets A and B to represent the sets above as follows:

We can also denote some sets that are infinite in size by using roster form.

Example 2: Match each set given below with its corresponding name.

1. {1, 2, 3, 4, …} A. Integers

2. {…-3, -2, -1, 0, 1, 2, 3, …} B. Whole Numbers

3. {0, 1, 2, 3, 4, …} C. Natural Numbers

You have seen that sets can even have ordered pairs of numbers as elements.

Example 3: On the axes shown to the right, a quadratic-linear system is graphed. State the solution set of this system in roster form.

The ________ of a set is a “part” of that set. All of the elements of the subset are members of the set itself. The _____________ (set with no elements) is a subset of every set. The ____________ (or ________) is denoted by the symbol { } or the symbol ∅ . In addition, any set is considered to be a subset of itself.

Example 4: Consider the set A defined as A = {1, 2, 3}. List all 8 subsets of A.

Suppose that set B = {1, 2, 3, 4, 5, 6} for a certain situation. Suppose, further, that set A = {2, 4, 6}. Then we know that A is a subset of B. We express that relationship with symbols: _____

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolThe _____________ of set A here is {1, 3, 5} because those are the elements in the set B that are ______ in set A. In other words, the complement of a subset consists of all elements in the set that are not in the subset.

Example 5: Consider the set E defined as E = {2, 3, 5, 7, 11, 13} and set F defined as F = {2, 5, 7}. Determine the complement of set F within set E. (We denote the complement of set F using many different symbols: ~F, F’, or Fc)

Example 6: Given the set of numbers on a standard six sided die, which of the following represents the complement of the set comprised of all multiples of 3?

(a) {1, 2, 3} (b) {2, 4, 6} (c) {1, 2, 3, 4, 5} (d) {1, 2, 4, 5}

Example 7: Suppose we have the set of real numbers. Graph the complement of each set.

We do not “add” two sets together. Nor do we subtract, multiply, or divide them. However, there are two operations on sets that are commonly used. We call the operations ________ and _____________. Just as when we use basic operations on numbers we get numbers for answers, when we operate on sets, we get sets for answers. Thus,

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High Schoolthe union of two sets is a set, and the intersection of two sets is a set. The definitions of union and intersection are given below in set-builder notation.

The union of two sets can be thought of as the “joining” of the sets (any element in either set must appear in their union). The intersection of two sets consists of whatever elements the two sets have in common.

Example 8: For each of the following, sets A and B are given. Find A ∪ B and A ∩ B.

Example 9: For each of the following, the graphs of two sets, A and B, are shown below. In each case, graph A ∪ B and A ∩ B.

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Example 10: For any set A, determine each of the following:

Example 11: Within the set of real numbers (-∞, 5] ∩ (-2, 7] = ?

(a) (-2, 7)(b) (-2, ∞)(c) [-2, 5](d) (-2, 5]

Many problem situations require extensive logic to sort through. A _________________ is a visual aid used to assist us in sorting through the logic of certain types of questions. Venn diagrams show the relationships between different sets.

5

Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolExample 12: Professor Boyette was trying to determine who was in her Advanced Statistics course. The Venn diagram below shows how she classified all of her students into different subsets. Based on this diagram, answer the following questions.

(a) How many of the students were female graduate students?

(b) How many total students are in the class?

(c) How many of Dr. Boyette’s students are male?

Example 13: A soft-drink company wanted to see which of two new drinks its consumers would prefer. To find out, the company surveyed 400 people who had tried both new drinks. 254 of the people surveyed liked drink A, 136 liked both drinks A and B, and 42 people liked neither. Which of the two drinks was preferred by the larger number of people?

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolPractice:

1. Central High School has 480 freshmen. Of those freshmen, 333 take Algebra, 306 take Biology, and 188 take both Algebra and Biology. How many freshmen at Central High School take neither Algebra nor Biology.

2. Eve was doing a science fair project by surveying her biology class. She found that of the 30 students in the class, 15 had brown hair and 17 had blue eyes and 6 had neither brown hair nor blue eyes. Which of the following represents the number of students who had brown hair and blue eyes?(a) 6 (b) 2 (c) 3 (d) 8

3. Use the Venn diagram below to answer each of the following:

Homework #11. In each of the following, a verbal description of a set is given. Write the elements

of the set using roster notation. (a) The set of all months that begin with the letter A. (b) The set of all days of the week that begin with T or S.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School(c) The set of all whole numbers that are greater than 6 and less than 11. (d) The set of all negative integers that are greater than -5. (e) The set of all factors of 18.

2. The graph below represents a set of points. Write the set in roster form.

3. The result of a statistical survey claims that 48% of cell phone users carry ACME Telecom as their cellular phone provider. The people that ran the survey state that results are good with a margin of error of 4%. The set of all possible percentages of people that carry ACME Telecom as their cellular phone provider according to this survey is:(a) {x: 48% < x < 54%}(b) {x: 46% < x < 50%}(c) {x: x < 44% or x > 52%}(d) {x: 44% < x < 52%}

4. The set {x: -10 < x < 8} can be written in interval notation as(a) [-10, 8) (b) [-10, 8] (c) (-10, 8) (d) (-10, 8]

5. List all subsets of the set A defined as A = {H, T}

6. In each case below, assume that the given universe is the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.(a) If A = {0, 2, 4, 6, 8}, then find A’. (b) If C = {1, 2, 4, 7, 8, 9}, then find Cc.

7. State whether each of the following or true or false. (If false, explain why.)(a) If A B and B C, then A C.

(b) The complement of the set of odd integers is the set of even integers.

8


(c) The complement of the set of all real numbers that are less than 5 is the set of all real numbers that are greater than 5.

8. In the set of real numbers, the complement of the set [5, ∞) would be which of the following?(a) (5, ∞) (b) (-5, ∞) (c) (-∞, 5) (d) (-∞, 5]

9. Complete the following table:

10. If A = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} and B = {(4, 7), (3, 5), (2, 3), (1, 2), (0, -1), (-1, -3)}, then find A ∩ B. (Hint: Your answer should consist of one or more ordered pairs.)

11. Use the Venn diagram below to answer each of the following.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School12. The accompanying diagram shows the number of students who own desktop

computers, laptop computers, and graphing calculators at a certain school. What percentage of students owns a laptop or a graphing calculator?

13. A high school surveyed 260 of its students regarding participation in either track or cross country. The result of the survey showed that 92 students participated in track only, 38 students participated in cross country only, and 86 students participated in neither sport. Of those surveyed, how many students participated in both track and cross country?

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolDay 2: Counting!Warm-Up

Peter throws a dice and spins a coin 150 times as part of an experiment. He records 71 heads and a six 21 times. On 68 occasions, he gets neither a head nor a six. Complete the table.

6 Not a six TotalHeadTail

TotalWhat percentage of times did Peter get both a head and a six?

The members of a tennis club are classified by gender and whether or not they are over 18.

o 12% are under 18 and femaleo 53% are over 18 and maleo 60% are male

Design and complete a two-way table to show this information using percentages in the blocks. What percentage of the members are under 18?

Total

TotalWhat percentage of the members are under 18?

So far we have counted the number of elements in a given set by using Venn Diagrams and other tools. We now would like to start to count _____________ of _____________. An experiment is any single stage or multi-stage process that has definable results. Those results form a set called the _______________ of the experiment. Today we will develop ways to count the number of elements in this sample space.

Example 1: Sundae Palace offers five flavors of ice cream: vanilla, chocolate, strawberry, peach and raspberry. A sundae can be made with either fudge or caramel topping, but not both. A sundae will be made that consists of one scoop of ice cream and one topping.

(a) What is the experiment?

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School(b) What is an outcome of this experiment?

(c) How many outcomes (sundaes) are there in this experiment’s sample space?

Tree Diagram Approach: Ordered Pair Approach:

Example 2: A coin is tossed three times in a row. Represent all outcomes in the sample space of this experiment by creating a tree diagram and a list of ordered triples.

Tree Diagram Approach: Ordered Triples:

In many practical situations, making a tree diagram or listing ordered pairs or triples is not practical. We can oftentimes shortcut this process by using the Fundamental

Counting Principle.

Example 3: Suppose a store offered 10 different types of work shirts and 12 different types of ties. How many different shirt/tie combinations are possible?

Example 4: Michael, Jacob, and Freddy are running a race in which there can be no ties. How many different orders can they finish in?

Example 5: California is trying devise a new license plate system. They want to use 7 characters to create their license plates. Determine how many plates are possible if the following conditions are imposed. There are 26 letters and 10 digits that can be used.

(a) The first three characters must be letters and the last four must be digits. Repetition of numbers and letters is allowed.

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(b) The first three characters must be letters and the last four must be digits. Repetition of numbers and letters is not allowed.

(c) The first three characters must be letters and the last four must be digits. The letters may repeat but the numbers may not.

The idea of rearranging objects is an important one in mathematics. We call such rearrangements permutations. The formal definition is given below:

Example 6: List all possible permutations of the letters given in the following sets. Count the total number of permutations for each.

(a) {B, A} (b) {B, A, T} (c) {B, A, T, H}

Clearly there must be a relationship between the number of elements of the set and the number of permutations of all of the elements. We will explore that relationship in the next example by using the Fundamental Counting Principle.

Example 7: Paul has chosen 10 songs for his playlist and claims that there are millions of different ways for him to order the songs. Is he correct?

When we want to count the number of permutations of all of the elements in a set we can use this approach. Because it is so common, there is a special notation for it called ____________.

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Example 8: Write out and evaluate each of the following without using your calculator. (a) 2! (b) 3! (c) 4! (d) 5!

Example 9: Noah rented 5 movies. Which of the following represents the total number of ways he can order them to be watched?

(a) 10 (b) 120 (c) 50 (d) 200

Sometimes we want to __________ just a subset of a given set of elements instead of the whole set. This process is illustrated in the next example.

Example 10: Ten people enter a race in which there can be no ties. Different awards are given for first through third place. In how many different ways can these awards be given out?

Permuting subsets is common enough and important enough to have unique notation.

Example 11: A coach is trying to pick five players for distinctly different positions on a basketball team. He has 9 players that he can pick from.

(a) Represent the number of different ways that the coach can fill these positions using permutation notation.

(b) Evaluate the number of different ways the coach can fill these positions using the Fundamental Counting Principle.

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(c) Evaluate the number of different ways the coach can fill these positions using the permutation option on your calculator.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolHomework #2

1. Franklin is taking a 3 question True/False quiz. After the quiz, Franklin is trying to determine all of the different ways he could have answered the three questions. For instance, the ordered triple (T, F, T) represents Franklin answering Question 1 with a true, Question 2 with a false, and Question 3 with a true. Using either a tree diagram or a list of ordered triples, determine the total number of ways Franklin could have answered these 3 questions.

2. A dinner menu lists two soups, seven meats, and three desserts. How many different meals consisting of one soup, two meats, and one dessert are possible? Assume that you cannot repeat your choice of meat.

3. Options on a bicycle include two types of handlebars, two types of seats, and a choice of 15 colors. How many possible versions of the bike are possible if a person must chose the handlebars, the seat and the color?

4. Consider the word JUNIOR.(a) How many different 4 letter arrangements are there of these letters if the

first letter is a J and there is no repetition?

(b) How many different 4 letter arrangements are there of these letters if the first letter is J or an N and there is no repetition?

5. Write out the multiplication form of 6! and find its value.

6. How many ways are there of rearranging all the letters in the word DEVIL?

7. Evaluate each of the following:(a) 7P4 (b) 8P3 (c) 9P2 (d) 10P1

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8. How many different 3-letter words can be formed from the letters in the word NUMERAL if there is no repetition allowed?(a) 21 (b) 500 (c) 210 (d) 5040

9. Marianna must create a 3-digit code from the numbers in the set {1, 2, 3, 5, 7}, where none of the 3-digits can be repeated. (a) Represent the total number of choices she has for her code in permutation

notation.

(b) Ashten believes he can find Marianna’s code by trying all the possibilities from part (a). If it takes him seconds to try a code, can he go through all the codes in less than 5 minutes?

10. In a telephone survey of 200 households, 108 households purchased wheat bread and 136 households purchased white bread. If 54 households purchased both types of bread, then how many of the households surveyed did not purchase either brand?

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolDay 3: Basic Probability Warm-Up

Complete the two-way table and use it to finda) How many went on holiday in August

b) How many stayed in a hotel in September

Hote

l Caravan Campin

gOther Tota

lJuly 11 4 3

August 14 6Septembe

r4 3 30

Total 49 15 11 100What was the survey about for this table?

Research was carried out on people traveling by train and by bus.o 88 out of 120 train travelers had made journeys of over 50 miles.o Of the 300 people questioned, 205 had travelled under 50 miles.

Draw a two-way table and use it to calculate the number of people who made a bus journey of under 50 miles.

Total

Total

Probability is one of the most important concepts in all of mathematics. At its core, probability is a measurement of how

likely an event, E, is to happen. In this lesson, we will review some of the most basic probability concepts that you have seen before…Example 1: Given that a standard die is rolled once, find the probability for each of the following events.

(a) P(rolling a 4) (b) P(rolling less than a 5) (c) P(rolling a 10)

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Example 2: Consider an event, E.

(a) If P(E) = 0, how do we interpret this probability?

(b) If P(E) = 1, how do we interpret this probability?

(c) Which of the following represents all possible probabilities for an event E?(a) 0 < P(E) < 100(b) 0 < P(E) < 1(c) 0 < P(E) < 1(d) -1 < P(E) < 1

Example 3: A bag contains eight geometric shapes: two squares, one rhombus, two scalene triangles, and three isosceles trapezoids. If one shape is pulled out at random, what is the probability all of its sides have equal lengths?

Card Problems!Some of the common probability problems center on decks of cards. Here is

a review of the cards in a standard deck:

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Example 4: Answer each of the following problems based on the cards in a standard deck.

(a) Given a standard deck of cards, if one card is drawn at random what is the probability that it will be a red queen?

(b) Given a standard deck of cards, if one card is drawn at random, what is the probability that it will be a black face card?

In every experiment in which an event E occurs, all outcomes in the sample space that are not in E are contained within the complement of E. This is the same idea as the complement of a set that we encountered earlier.

Example 5: Consider rolling a standard die once. (a) What is the probability that the number rolled is less than 3?

(b) What is the probability that the number rolled is not less than 3?

(c) In general, what is the sum P(E) + P(not E) equal to?Example 6: If the chance of Jenna bringing a cheese sandwich to school for lunch is 55%, then what is the probability that she will not bring a cheese sandwich to school?

Probability can start to get tricky when we combine two or more activities or “events” and try to determine probability from them. Often, these problems

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High Schoolcan be best analyzed by constructing diagrams or listing outcomes that lie

in the experiment’s sample space.

Example 7: A fair coin is tossed twice and the outcome is recorded each time.

(a) Construct a tree diagram that represents all outcomes in this experiment’s sample space.

(b) Create a list of ordered pairs that represents all outcomes in this experiment’s sample space.

(c) Using either (a) or (b), determine the probability of getting exactly one head.

Example 8: A family moves next door. You know that they have three children, but not whether they are boys or girls.

(a) Draw a tree diagram that represents all possible gender combinations for the three children.

(b) List the ordered triples that coincide with the diagram you drew in part (a).

(c) What is the probability that the family next door has two girls and one boy?

(d) What is the probability that the family next door has all girls or all boys?Example 9: Franco has 2 quarters, 2 dimes and 1 nickel in his pocket. He pulls two coins out at random without replacement.

(a) Construct a tree diagram to represent all possible outcomes of the coins that Franco could have pulled out of his pocket.

(b) What is the probability he has less than 35 cents in his hand?

Example 10: Lacy has four good friends, three females (Jennifer, Eve, and Carla) and one male (Franklin). She is taking two trips to Florida and can take one friend each time, who she will randomly choose.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School(a) If the same friend will not be chosen for each trip, construct a tree diagram or a

set of ordered pairs that represent all possible outcomes for whom Lacy will take on the trip.

(b) Determine the probability that Lacy will take females both times.

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1. A standard fair die is tossed. Find the probability that the number rolled is:(a) 4 (b) even (c) less than 2 (d) 9

(e) prime (f) odd (g) greater than 2(h) not greater than 5

2. One card is chosen at random from a standard deck of cards. Find the probability of the following:

(a) P(Red) (b) P(face card) (c) P(7) (d) P(a diamond)

(e)P(red 8) (f) P(king) (g) P(not a heart) (h) P(not a red 2)

3. The probability that Mr. Ford is going to buy a new car is 3/42. What is the probability that Mr. Ford will not buy a new car?

4. The local weather station predicts a 35% chance of snow tomorrow. Which of the following represents the probability that it will not snow?

(a) 85% (b) 65% (c) 35% (d) 55%

5. If a coin is tossed twice in a row, one outcome is getting two heads. Is getting two tails the complement of this event? Explain.

6. When a coin and a standard 6-sided die are tossed simultaneously, the number of outcomes in the sample space is:

(a) 8 (b) 2 (c) 12 (d) 36

7. A spinner shows 3 regions numbered {1, 2, 3}, all equally likely to occur when the spinner is spun. When the arrow is spun twice, the number of pairs in the outcome set is:

(a) 6 (b) 2 (c) 3 (d) 9

8. Two coins and a standard die are tossed simultaneously. The number of outcomes in the sample space is:

(a) 10 (b) 24 (c) 3 (d) 8

9. A fair coin is tossed three times in a row and the outcome is noted each time.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School(a) Draw a tree diagram that

represents all possible outcomes for this experiment.

(b) Create a list of all the ordered triples that represent the outcomes of this experiment.

(c) Find the probability that you get exactly two heads.

(d) Find the probability that all the coins are heads.

10. To navigate a maze, a mouse must make 4 turns, which can be either LEFT or RIGHT. (a) Draw a tree diagram to represent all the possible paths that the mouse could

take.

(b) Find the probability the mouse takes all lefts.

(c) In order to reach the exit, the mouse must take 3 rights and one left, in any order. What is the probability the mouse reaches the exit?

Day 4: Independent vs. Dependent Events Warm-Up

Highest Level of Educational AchievementPrimary News

SourceNot High School

GraduateHS Graduate but Not

CollegeCollege

GraduateTota

lNewspapers 49 205 188 442

Local Television 90 170 75 335Cable Television 113 496 147 756

Internet 41 401 245 68724

Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolNone 77 165 38 280Total 370 1,437 693 2,50

0

1. What is the probability that a person’s primary news source is the internet?

2. What is the probability that a person is a college graduate?

3. What is the probability that a person’s primary news source is the internet OR they are a college graduate?

4. What is the probability that a person’s primary news source is the internet AND they are a college graduate?

5. What is the probability that a college graduate’s primary news source is the internet?

6. What is the probability that someone who uses the internet as their primary news source is a college graduate?

7. Are the events “uses internet as primary news source” and “college graduate” independent? Justify.

In our last lesson, we learned how to solve more complex probability problems by using diagrams or lists to visualize the sample space of an experiment. Today we will find probabilities without creating these diagrams. We will concentrate first on what are known as independent events—events where the probability of one event is not affected by the other event occurring.

Example 1: Consider tossing a fair coin and then rolling a fair six-sided die. (a) Why are these two events independent?

(b) Draw a tree diagram to represent all the possible outcomes from this experiment.

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(c) What is the probability of getting a tails when tossing the coin?

(d) What is the probability of rolling a number less than three on the die?

(e) From your tree diagram, what is the probability that a tail was tossed and a number less than three was rolled?

(f) How could the probabilities in parts (c) and (d) be combined to obtain the probability found in part (e)?

Example 2: A fair six-sided die is rolled two times. What is the probability that it will land on a four each time?

The key to many of these problems is to reword the problem so that the “AND” nature of it becomes apparent.

Example 3: Michael flipped a fair coin three times and in each case it came up heads. Which of the following represents the probability that this will happen?

(a) 2/3 (b) ¼ (c) ½ (d) 1/8

Example 4: Ava is taking a multiple choice quiz for which she has not studied. There are 5 questions on the quiz and each question has four possible choices. If Ava guesses on each of the questions, what is the probability that she will receive a perfect score?

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Example 5: Jack consistently makes 80% on all of his free throws in basketball. Which of the following represents the probability that he will miss two free throws in a row?

(a) 16% (b) 56% (c) 40% (d) 4%

Example 6: The weather forecast states that there is a 40% chance of rain on Saturday and a 70% chance of rain on Sunday.

(a) What is the probability that it will rain on Saturday and Sunday?

(b) What is the probability that it will rain neither Saturday nor Sunday?

(c) Why don’t the probabilities that you found in parts (a) and (b) add up to 1 (or 100%)?

________________events are any two events where the probability of the second event is affected by the result of the first event. Card probability

problems offer an excellent introduction to this idea.Example 7: A card is pulled from a standard deck and its result is noted. Then, without replacing the first card, a second card is drawn from the deck and its result is noted. Determine the following probabilities:

(a) Two kings are drawn (b) Two red cards are drawn (c) Two face cards are drawn

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Example 8: A jar containing 8 red marbles and 12 green marbles is on Mr. Cook’s desk. A marble is chosen at random, not replaced, and another marble is chosen. What is the probability that the following outcome occurs?

(a) A red marble first, then a green marble

(b) Two red marbles

(c) Two green marbles

(d) A red marble, then a blue marble

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolExample 9: Two cards are chosen at random from a standard deck of cards without replacement. Which of the following represents the probability that two aces are drawn?

(a) 1/26 (b) 7/103 (c) 1/221 (d) 1/169

Example 10: Eight runners enter a race—four from Sanderson High School and four from Millbrook High School. If the runners finish the race at random, what is the probability that Sanderson runners finish in first, second, and third place?

Example 11: Zoe is talking with Peter about her plans this weekend. She tells Peter that if it doesn’t rain on Saturday then there is a 30% probability she will spend the day at the beach and if it does rain, she won’t go at all. The weather forecast states that the probability of rain on Saturday is 20%. Which of the following is the probability that Zoe will go to the beach?

(a) 50% (b) 6% (c) 36% (d) 24%

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1. Which of the following gives the probability that a fair coin will land heads up four times in a row?

(a) 1/16 (b) 1/8 (c) 1/2 (d) 1/64

2. A coin is tossed 5 times and a tail is the outcome each time. What is the probability of getting a tail on the sixth throw?

(a) 1/64 (b) 1/32 (c) 1/6 (d) ½

3. A fair coin is tossed and a fair die is rolled. Which of the following gives the probability that a multiple of two will be rolled on the die and a head will be tossed on the coin?

(a) ¼ (b) ½ (c) 2/3 (d) ¾

4. The probability of snow for any given weekend in December is 60%. Which of the following is closest to the probability of getting snow four consecutive weekends in December?

(a) 24% (b) 35% (c) 13% (d) 2%

5. The probability it will rain this Saturday is 20% and the probability it will rain this Sunday is 30%. What is the probability that it won’t rain this weekend?

(a) 56% (b) 94% (c) 50% (d) 36%

6. If Mark has a probability of 30% of bowling a strike, then answer the following questions:(a) What is the probability that he bowls 2 strikes in a row? Give your answer as

a percent.

(b) What is the probability that he doesn’t bowl a strike 10 times in a row? Give your answer as a percent to the nearest tenth of a percent.

7. At the local fair a game is created where a person tosses three bean bags at a board as shown below. A player wins if all three bean bags fall in the white squares. (a) Find the probability that a player will win the game.

(b) If Mark plays the game five times, find the probability that he will lose all five games. Express your answer as a percent accurate to the nearest tenth. HINT: Using part (a), determine the probability Mark will lose a single game.

8. Which of the following is the probability that if two cards are chosen at random from a standard deck of cards without replacement they are both red?

(a) 1/26 (b) 51/103 (c) 25/102 (d) 5/26

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9. Mr. Edgar is choosing students from his French club to be president and vice-president. The club consists of 12 girls and 7 boys. If Mr. Edgar pulls names randomly from a hat, without replacement, what is the probability that a girl is chosen for both the president and vice-president positions?

(a) 22/57 (b) 1/6 (c) 7/22 (d) 7/12

10. In a one-on-one free throw shooting situation in basketball, a player gets to shoot a second free throw if he or she has made the first one already. If Fiona has a 70% chance of making any given free throw, in one-on-one situation, what is the probability she will miss the second free throw? HINT: In order to miss the second free throw she must have made the first one.

11. A bowl of mints is placed by the cashier at the diner. The mints consist of 9 pink mints and 5 yellow mints. If Aaron randomly reaches in and grabs two mints, without replacement, what is the probability he will grab:(a)Two pink mints (b) two yellow mints (c) both mints the same

color

12.

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolDay 5: Mutually Exclusive EventsWarm-Up

1. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event.

____A. The probability of this is impossible. It can never occur. ____B. This event is certain. It will occur on every trial of the

random phenomenon.____C. This event is very unlikely, but it will occur once in a

while in a long sequence of trials. ____D. This event will occur more often than not.

2. If P(B) = 0.4 and P(A ∩ B) = 0.21, then find P(A) if A and B are independent.

3. Suppose P(A) = 0.35, P(B) = 0.51 and P(A ∩ B) = 0.17. Find(a) P(A’) (b) P(A ∪ B) (c) Are A and B independent events?

Explain.

4. Which of the following are true?I. Two events are mutually exclusive if they can’t both occur at the same

time.II. Two events are independent if they have the same probability.III. An event and its complement have probabilities that always add to 1.

(a) I only (b) II only (c) I and II only (d) I and III only

5. Of voters in a recent election, 57% were male, 64% were Democrat, and 35% were both male and Democrat. Are being male or Democrat independent of each other? Justify your answer.

6. The probability that a student in Math II is a girl is 53%. The probability that a student plays a sport is 35%. The probability that a girl plays sports is 17%. Are these two events independent? Justify your answer.

Mutually exclusive events are two events which have ___________________. The probability that these two events would

occur at the same time is ____________.

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1. 02. 0.013. 0.34. 0.65. 0.996. 1


Example 1: A single card is drawn from a standard deck of playing cards.

Let A = The event of drawing a black queen Let B = The event of drawing a red five

(a) Why are A and B mutually exclusive events?

(b) What is P(A and B)?

Calculating probabilities involving mutually exclusive events is extremely important and fairly easy, as the next example will

illustrate.

Example 2: A fair six-sided die is rolled. What is the probability of rolling a number less than 3 or rolling a 5?

Example 3: 1 card is drawn at random from a standard deck. Find the probability the card is:

(a) A king or a queen (b) A king and a queen

The concept of mutually exclusive events can now allow us to solve harder probability problems that involve what we have seen before. The key will be to identify the mutually exclusive events that make up the larger event.

Example 4: Two cards are drawn at random from a deck of cards without replacement. What is the probability the two cards are a queen and a king?

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Example 5: Two standard dice are rolled. What is the probability the sum of the numbers on the dice is equal to 9?

Example 6: Faith has 6 quarters, 2 nickels, 1 dime, and 3 pennies in her coin purse. She pulls out two coins randomly without replacement. What is the probability Faith has at least 35 cents in her hand?

Non-mutually exclusive events are two events which _________ one or more of the same outcomes. The example below will illustrate how to think about these events and their associated probabilities. Example 7: A standard six-sided die is rolled once.

Let A = The event of rolling an even Let B = The event of rolling a multiple of 3

(a) List all of the elements in each of these events

A: B:

(b) Find the probability of A (c) Find the probability of B

(d) Find the probability of A or B (e) Why is the answer from part (d) not the

sum of the answers from parts (b) and (c)?

Example 8: If one card is drawn at random from a standard deck of cards, what is the probability of randomly selecting a 10 or a red card?

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Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High SchoolExample 9: If a fair six sided die is rolled, what is the probability of rolling an odd number or an number less than 4?

In simple situations involving non-mutually exclusive events, the formula shown at the top does not need to be used because outcomes can be counted

easily enough. Sometimes, though, this formula is necessary.

Example 10: On a given night in March, the probability it is going to rain is 0.60, the chance it is going to rain and snow is 0.17, and the chance it is going to snow is 0.32. What is the probability it is going to rain or snow?

Example 11: The probability the Fitzy rugby team will win is 68%. The probability a team member gets injured in the game is 36% and the probability a team member gets hurt or they win the game is 86%. What is the probability the Fitzy team wins and a team member gets hurt in the game?

Example 12: Shana would like to take her son Maxwell to the zoo one day this weekend but will only do so if it doesn’t rain. The probability of rain on Saturday is 20% and the probability of rain on Sunday is 40%. Assume the events of rain on Saturday and Sunday are independent of each other.

(a) What is the probability it won’t rain on Saturday?

(b) What is the probability it won’t rain on Sunday?

(c) What is the probability it won’t rain on Saturday and it won’t rain on Sunday?

(d) What is the probability it won’t rain either Saturday or Sunday?

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1. A person is shopping for a new snowmobile. The probability someone buys an Arctic Cat is 39%, a Polaris is 27% and a Ski-Doo is 18%. Which of the following is the probability this person buys either an Arctic Cat or a Ski-Doo?

(a) 84% (b) 66% (c) 57% (d) 45%

2. A single, fair 6-sided die is thrown. Which of the following is the probability it lands on a multiple of 2 or a five?

(a) 2/3 (b) 1/6 (c) 1/3 (d) 5/6

3. A pair of dice are thrown. Which of the following represents the probability a sum of 11 is thrown?

(a) 11/36 (b) 6/11 (c) 1/36 (d) 1/18

4. Two cards are drawn at random from a standard deck without replacement. Which of the following represents the probability the two cards are a give and a six (in either order)?

(a) 8/663 (b) 2/13 (c) 4/663 (d) 4/13

5. Two cards are drawn at random from a standard deck without replacement. Which of the following represents the probability the two cards drawn are either both kings or both queens?

(a) 2/13 (b) 1/221 (c) 2/221 (d) 8/13

6. A bag of marbles contains 8 red marbles and 6 yellow marbles. Two marbles are drawn out of the bag at random without replacement. What is the probability that

(a) first a red marble then a yellow marble are drawn?

(b) Two red marbles are drawn?

(c) Two yellow marbles are drawn?

(d) Two marbles of the same color are drawn?

7. A particular history class at Sanderson High School has the following breakdown of students by grad and by gender:

Grade Gender6 Freshmen 3 Girls and 3 Boys

16 Sophomores 10 Girls and 6 Boys

8 Juniors 5 Girls and 3 Boys36

Honors Math 2 Unit 7 Notes/Activities/HomeworkSanderson High School30 Total Students

18 Girls and 12 Boys

One student is chosen at random from the 30 total students to give a speech the next day. Find the probability the student chosen is:

(a) A girl (b) a sophomore (c) a girl or a sophomore

(d) Why is the probability you calculated in part (c) not the sum of the probabilities you found in parts (a) and (b)?

8. If one card is pulled from a standard deck of cards, which of the following is the probability the card is a red card or an ace?

(a) 15/52 (b) 30/52 (c) 28/52 (d) 26/52

9. If a child tossed a fair six-sided die, which of the following is the probability an odd number or a number greater than 3 would be showing?

(a) 5/6 (b) 2/3 (c) 1/3 (d) 1/6

10. The probability Emma orders French fries at lunch is 0.32 and the probability she orders a grilled cheese sandwich and fries is 0.65. If the probability she orders just a grilled cheese sandwich is 0.76, what is the probability she will order a grilled cheese or fries?

11. If a person is picked at random from the general population there is a 0.52 probability the person is a woman, a 0.56 probability the person is younger than 40 and a 0.78 probability the person is a woman or is younger than 40. What is the probability a person picked at random is a woman and younger than 40?

12. The probability it is going to rain on Saturday is 50% and the probability it is going to rain on Sunday is 80%. Assuming these two events are independent, find the following:

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(a) The probability it will rain on Saturday and Sunday.

(b) The probability it will rain Saturday or Sunday.

13. Two events A and B have probabilities given below:

P(A) = 1/3 P(B) = ½ P(A or B) = 5/6

Are events A and B mutually exclusive or non-mutually exclusive? Justify your answer.

HINT: Determine the probability of A and B.

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cferreras.weebly.com · Web viewThe _____ (set with no elements) is a subset of every set. The _____ (or _____) is denoted by the symbol { } or the symbol ∅ . In addition, any set

cferreras.weebly.com · Web viewThe ___ (set with no elements) is a subset of every set. The _ (or ___) is denoted by the symbol { } or the symbol ∅ . In addition, any set