Transcript Analytical Geometry - Bibb County Public School District
Unit 7 - Probability
Analytical Geometry
Bellringer 8/11/14
• What do you know about Probability?
The Probability Song (T.D. Version)
Today we’re going to learn about probabilities, and we’re going to make diagrams that look like trees.
We’re going to flip some coins so we can draw our trees, and we’re going to calculate our probabilities.
Probability’s the likelihood of a chance event. Hey!! Ho!! Hey!!
It’s between 0 and 1 and that’s just common sense. Hey!! Hey!!
An event is any outcome from an experiment.
If you can figure them all out, then you’re intelligent. (Repeat) Today we’re going to learn about probabilities, and we’re going to make diagrams that look like trees.
On the way to Probability….
What is Probability?
Probability is a
measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes. 1. The probability of a given event is a fraction between 0 and 1. 2. The sum of all probabilities of every outcome should always equal 1. 3.
What does it mean to have a probability of zero?
A probability of zero means that an event is impossible.
What does it mean to have a probability of One?
Question: A single 6-sided die is rolled. What is the probability of rolling a number less than 7? Answer: Rolling a number less than 7 is a certain event since a single die has 6 sides, numbered 1 through 6. A probability of one means that an event is certainly going to happen.
Why are probabilities between 0 and 1?
Probabilities are always between 0 and 1 because every event has a chance of occurrence that lies between 0% (no chance of happening, corresponds to 0) and 100% (certainty of happening, corresponds to 1).
Tree Diagrams
Tree diagram:
A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event.
• Here is a tree diagram for the toss of a coin:
Tree Diagram Create a tree diagram for the possible outcomes and probabilities for the two tosses. Probabilities The Sum of all Possible outcomes
Bellringer 8/12/14
• Kendra is playing a card game with a standard 52-card deck. She wants her first draw to be a heart or an ace.
1.
How many ways can Kendra draw a heart or an ace? 2.
How many ways can Kendra draw a card that is neither a heart nor an ace?
Introduction
Probability is a number from 0 to 1 inclusive or a percent from 0% to 100% inclusive that indicates how likely an event is to occur. In the study of probability, an experiment is any process or action that has observable results. The results are called outcomes. The set of all possible outcomes is called the sample space. An event is an outcome or set of outcomes of an experiment; therefore, an event is a subset of the sample space, meaning a set whose elements are in another set, the sample space. In this lesson, you will learn to describe events as subsets of the sample space. For example, drawing a card from a deck of cards is an experiment. All the cards in the deck are possible outcomes, and they comprise the sample space. The event that a jack of hearts is drawn is a single outcome; it is a set with one element: {jack of hearts}. An element is an item or a member in a set. The event that a red number card less than 5 is drawn is a set with eight elements: {ace of hearts, ace of diamonds, 2 of hearts, 2 of diamonds, 3 of hearts, 3 of diamonds, 4 of hearts, 4 of diamonds}. Each event is a subset of the sample space.
Note that the words “experiment” and “event” in probability have very specific meanings that are not always the same as the everyday meanings. In the example on the previous slide, “drawing a card” describes an experiment, not an event. “Drawing a jack of hearts” and “drawing a red number card less than 5” describe events, and those events are sets of outcomes.
The important distinction to remember is that an event is a set of one or more outcomes, while the action or process is the experiment.
Bellringer 8/13/14
1.
2.
What is the Sample Space of rolling 2 dice?
Write an event for the experiment mentioned in Question1.
Key Concepts
A set is a list or collection of items. Set A is a subset of set B, denoted by A C = {2, 4} ⊂ B, if all the elements of A are also in B. For example, if B = {1, 2, 3, x}, then some subsets of B are {1, 2, 3}, {2, x}, {3}, and {1, 2, 3, x}. Note that a set is a subset of itself.
An empty set is a set that has no elements. An empty set is denoted by . An empty set is also called a null set. Equal sets are sets with all the same elements. For example, consider sets A, B, and C as follows: A is the set of integers between 0 and 6 that are not odd. B is the set of even positive factors of 4. A = B = C because they all have the same elements, 2 and 4.
The union of sets A and B, denoted by , is the set of elements that are in either A or B or both A and B.
Bellringer 8/14/14
1) What is the probability of rolling an even number on a 6-sided die?
1) For the previous question write the set for the Sample space and the subset for the event.
Common Errors/Misconceptions
1) confusing the meanings of event and experiment 2) confusing union and intersection of sets 3) neglecting order, thereby neglecting to identify different outcomes such as HT and TH
Key Concepts, continued
An intersection is a set whose elements are each in both of two other sets. The intersection of sets A and B, denoted by , is the set of elements that are in both A and B.
For example, if A = {a, b, c} and B = {m, a, t, h}, then:
A A
È Ç
B B
= space. = { a, b, c, m, t, h {a} B, so the circles overlap: } The complement of set A, denoted by , is the set of elements that are in some universal set, but not in A. The complement of A is the event that does not occur.
Sometimes it is helpful to draw tables or diagrams to visualize outcomes and the relationships between events. A Venn diagram is a diagram that shows how two or more sets in a universal set are related. In this diagram, members in event A also fit the criteria for members in event
Bellringer 8/18/14
1) What is the Probability of your phone landing screen side down when you drop it?
2) How do you express Probability as a ratio?
**Turn in your Take home test to the appropriate current work folder for your class** 1 st – Green, 2 nd – Gray, 7 th - Blue
Probability words to know!
Probability:
a measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes. The probability of a given event is a fraction between 0 and 1. The sum of all probabilities of every outcome should always equal 1. • Experiment: a process or ACTION with an observable result.
• Outcomes: the observable results of an experiment.
• Sample Space: the set of ALL possible outcomes of an experiment.
• Event: an outcome or set of outcomes of an experiment.
• Subset: a set whose elements are all in another set.
Need to know about Sets!
• Set: a list or collection of items • Subset – denoted by B ⊂ , as in A ⊂ B where A is a subset of B, where all of the elements of A are also in • Empty set: also known as a null set, is a set with no elements denoted by ∅ • Equal sets: sets with the exact same elements • Union of Sets: joining of all of the elements of two sets denoted by 𝐴 ∪ 𝐵 , where A and B are being combined • Intersection of sets: the elements that are alike or shared by two sets denoted by and B 𝐴 ∩ 𝐵 , the intersection of A and B contains the elements that are in both A • Complement of a set: the set of elements that are in some universal set, but not in set A,
denoted by Ā or A’
Common Errors/Misconceptions
• confusing the meanings of event and experiment • confusing union and intersection of sets • neglecting order, thereby neglecting to identify different outcomes such as HT and TH
Guided Practice
• • •
Example 1
Hector has entered the following names in the contact list of his new cell phone: Alicia, Brisa, Steve, Don, and Ellis. He chooses one of the names at random to call. Consider the following events. B: The name begins with a vowel. • E: The name ends with a vowel. • Draw a Venn diagram to show the sample space and the events B and E. Then describe each of the following events by listing outcomes. B E 𝐵 ∪ 𝐸 𝐵 ∩ 𝐸 𝐵 ∪ 𝐸
Guided Practice: Example 1, continued 1.
Draw a Venn diagram. Use a rectangle for the sample space. Use circles or elliptical shapes for the events B and E.
Write the students’ names in the appropriate sections to show what events they are in. 7.1.1: Describing Events
23
Guided Practice: Example 1, continued 2.
List the outcomes of B.
B
= {Ellis, Alicia} 7.1.1: Describing Events
24
Guided Practice: Example 1, continued 3.
List the outcomes of E.
E
= {Alicia, Brisa, Steve} 7.1.1: Describing Events
25
Guided Practice: Example 1, continued 4.
List the outcomes of
B
= {Ellis, Alicia}
E
= {Alicia, Brisa, Steve}
B
Ç
E
is the intersection of events
B
and
E
. Identify the outcome(s) common to both events.
B
Ç
E
= {Alicia} 7.1.1: Describing Events
26
Guided Practice: Example 1, continued 5.
List the outcomes of
B
= {Ellis, Alicia}
E
= {Alicia, Brisa, Steve}
B
È
E
is the union of events
B
and
E
. Identify the outcomes that appear in either event or both events.
B
È
E
= {Ellis, Alicia, Brisa, Steve} 7.1.1: Describing Events
27
Guided Practice: Example 1, continued 6.
B
B
= {Ellis, Alicia} Sample space = {Alicia, Brisa, Steve, Don, Ellis}
B
is the set of all outcomes that are in the sample space, but not in
B
.
B
= {Brisa, Steve, Don} 7.1.1: Describing Events
28
Guided Practice: Example 1, continued 7.
List the outcomes of .
B
= {Ellis, Alicia}
E
= {Alicia, Brisa, Steve}
B
È
E
= {Ellis, Alicia, Brisa, Steve} Sample space = {Alicia, Brisa, Steve, Don, Ellis}
B
È
E
is the set of all outcomes that are in the sample space, but not in
B
È
E
. ✔
B
È
E
= {Don} 7.1.1: Describing Events
29
Guided Practice
•
Example 2
• An experiment consists of rolling a pair of dice. How many ways can you roll the dice so that the product of the two numbers rolled is less than their sum?
Guided Practice: Example 2, continued 1.
Begin by showing the sample space.
This diagram of ordered pairs shows the sample space.
(1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6) Key: (2, 3) means 2 on the first die and 3 on the second die.
7.1.1: Describing Events
31
Guided Practice: Example 2, continued 2.
Identify all the outcomes in the event “the product is less than the sum.”
For (1, 1), the product is 1 × 1 = 1 and the sum is 1 + 1 = 2, so the product is less than the sum. For (1, 2), the product is 1 × 2 = 2 and the sum is 1 + 2 = 3, so the product is less than the sum. The tables on the next two slides show all the possible products and sums. 7.1.1: Describing Events
32
Guided Practice: Example 2, continued Outcome Product Sum Product < sum Outcome Product Sum Product < sum
(1, 1) 1 2 Yes (2, 4) 8 6 No (1, 2) 2 3 Yes (2, 5) 10 7 No (1, 3) 3 4 Yes (2, 6) 12 8 No (1, 4) 4 5 Yes (3, 1) 3 4 Yes (1, 5) 5 6 Yes (3, 2) 6 5 No (1, 6) (2, 1) 6 2 7 3 Yes Yes (3, 3) (3, 4) 9 12 6 7 No No (2, 2) (2, 3) 4 6 4 5 No No (3, 5) (3, 6) 15 18 8 9 No No
(continued)
7.1.1: Describing Events
33
Guided Practice: Example 2, continued Outcome Product Sum Product < sum Outcome Product Sum Product < sum
(4, 1) 4 5 Yes (5, 4) 20 9 No (4, 2) 8 6 No (5, 5) 25 10 No (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) 12 16 20 24 5 10 15 7 8 9 10 6 7 8 No No No No Yes No No (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) 30 6 12 18 24 30 36 11 7 8 9 10 11 12 No Yes No No No No No 7.1.1: Describing Events
34
Guided Practice: Example 2, continued
By checking all the outcomes in the sample space, you can verify that the product is less than the sum for only these outcomes: (1, 1) (2, 1) (1, 2) (3, 1) (1, 3) (4, 1) (1, 4) (5, 1) (1, 5) (6, 1) (1, 6) 7.1.1: Describing Events
35
Guided Practice: Example 2, continued 3.
Count the outcomes that meet the event criteria.
There are 11 ways to roll two dice so that the product is less than the sum. ✔ 7.1.1: Describing Events
36
Bellringer 8/19/14
1.
Create an experiment with 2 events.
2.
Draw a Venn Diagram to illustrate the experiment with the two events.
3.
Put away your notebooks and other items. I should only see a pencil and eraser on your desk!
• • •
Key Concepts
The conditional probability of B given A is the probability that event B occurs, given that event A has already occurred. If A and B are two events from a sample space with P(A) ≠ 0, then the conditional probability of B given A, denoted ( ) , has two equivalent expressions: ( ) = ( and
B
) = number of outcomes in (
A
and
B
number of outcomes in
A
) • • • • The second formula can be rewritten as Using set notation, conditional probability is written like this: That is why the formula for ( ) ( ) ,is read “the probability of B given A.” ( ) The “conditional probability of B given A” only has meaning if event A has occurred. has the requirement that P(A)≠0. = ( Ç ( )
B
) The conditional probability formula can be solved to obtain a formula for P(A and B), as shown on the next slide.
• • • • • ( ) ( ) ( ) • • =
P B A
( ( ) ( ) and
P A
= = ( (
B
) and ( ) and
B B
) ) ( and
B
) = ( ) • ( ) • ( ) Write the conditional probability formula.
Multiply both sides by
P
(
A
). Simplify. Reverse the left and right sides.
Remember that independent events are two events such that the probability of both events occurring is equal to the product of the individual probabilities. Two events A and B are independent if and only if ( Ç
B
) = ( ) •
P B
. The occurrence or non-occurrence of one event has no effect on the probability of the other event. If A and B are independent, then the formula for P(A and B) is the equation used in the definition of independent events, as shown below.
( and
B
( and
B
) ) = = ( ) ( ) • • ( )
P B
formula for
P
(
A
and
B
) formula for
P
(
A
and
B
) if
A
and
B
are independent
• •
Key Concepts, continued
The following statements are equivalent. In other words, if any one of them is true, then the others are all true. • • • • Events A and B are independent. The occurrence of A has no effect on the probability of B; that is, The occurrence of B has no effect on the probability of A; that is, ( ) ( ) = = P(A and B) = P(A) • P(B).
( ) .
( ) .
• Note: For real-world data, these modified tests for independence are sometimes used: ( ) » ( ) .
• • Events A and B are independent if the occurrence of A has no significant effect on the probability of B; that is, ( ) » ( ) .
Events A and B are independent if the occurrence of B has no significant effect on the probability of A; that is, • When using these modified tests, good judgment must be used when deciding whether the probabilities are close enough to conclude that the events are independent.
Bellringer 8/20/14
1.
Define Probability
2.
Define Experiment
3.
Define Event
4.
Define Outcome
•
Guided Practice
•
Example 1
• Alexis rolls a pair of number cubes. What is the probability that both numbers are odd if their sum is 6? Interpret your answer in terms of a uniform probability model.
Guided Practice: Example 1, continued 1.
Assign variable names to the events and state what you need to find, using conditional probability.
Let
A
be the event “Both numbers are odd.” Let
B
be the event “The sum of the numbers is 6.” You need to find the probability of
A
given
B
. That is, you need to find ( ) .
7.2.1: Introducing Conditional Probability
43
Guided Practice: Example 1, continued 2.
Show the sample space.
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6) Key: (2, 3) means 2 on the first cube and 3 on the second cube.
7.2.1: Introducing Conditional Probability
44
Guided Practice: Example 1, continued 3.
Identify the outcomes in the events.
The outcomes for
A
are
bold and purple
.
A: Both numbers are odd.
(1, 1)
(2, 1)
(3, 1)
(1, 2) (2, 2) (3, 2)
(1, 3)
(2, 3)
(3, 3)
(1, 4) (2, 4) (3, 4)
(1, 5)
(2, 5)
(3, 5)
(1, 6) (2, 6) (3, 6) (4, 1)
(5, 1)
(6, 1) (4, 2) (5, 2) (6, 2) (4, 3) (4, 4)
(5, 3)
(5, 4) (6, 3) (6, 4) (4, 5) (4, 6)
(5, 5)
(5, 6) (6, 5) (6, 6) 7.2.1: Introducing Conditional Probability
45
Guided Practice: Example 1, continued
The outcomes for
B
are
bold and purple
.
B: The sum of the numbers is 6.
(1, 1) (1, 2) (1, 3) (1, 4)
(1, 5)
(1, 6) (2, 1) (2, 2) (2, 3)
(2, 4)
(2, 5) (2, 6) (3, 1) (3, 2)
(3, 3)
(3, 4) (3, 5) (3, 6) (4, 1)
(4, 2)
(4, 3) (4, 4) (4, 5) (4, 6)
(5, 1)
(5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) 7.2.1: Introducing Conditional Probability
46
Guided Practice: Example 1, continued 4.
Identify the outcomes in the events and B.
Use the conditional probability formula: = ( Ç ( )
B
) .
A
Ç
B A
Ç = the outcomes that are in
A
and also in
B
.
B
= { ( ) , 3, 3 , 5, 1 }
B
= {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} 7.2.1: Introducing Conditional Probability
47
Guided Practice: Example 1, continued 5.
Find and
A
Ç
B
has 3 outcomes; the sample space has 36 outcomes. ( Ç
B
) = number of outcomes in
A
and
B
number of outcomes in sample space = 3 36
B
has 5 outcomes; the sample space has 36 outcomes.
( ) = number of outcomes in
B
number of outcomes in sample space = 5 36 7.2.1: Introducing Conditional Probability
48
Guided Practice: Example 1, continued 6.
Find
= ( Ç ( )
B
) = 3 36 5 36 ( ) = 3 36 • 36 5 Write the conditional probability formula.
Substitute the probabilities found in step 5.
To divide by a fraction, multiply by its reciprocal. 7.2.1: Introducing Conditional Probability
49
Guided Practice: Example 1, continued
= 3 1 36 • 36 1 5 ( ) = 3 5 Simplify. 7.2.1: Introducing Conditional Probability
50
Guided Practice: Example 1, continued 7.
Verify your answer.
Use this alternate conditional probability formula: ( ) = number of outcomes in
A
Ç number of outcomes in
B B
.
A
Ç
B
has 3 outcomes: {(1, 5), (3, 3), (5, 1)}.
B
has 5 outcomes: {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}. ( ) = number of outcomes in
A
Ç
B
number of outcomes in
B
( ) = 3 5 7.2.1: Introducing Conditional Probability
51
Guided Practice: Example 1, continued 8.
Interpret your answer in terms of a uniform probability model.
The probabilities used in solving the problem are found by using these two ratios: number of outcomes in an event number of outcomes in the sample space number of outcomes in an event number of outcomes in a subset of the sample space 7.2.1: Introducing Conditional Probability
52
Guided Practice: Example 1, continued
These ratios are uniform probability models if all outcomes in the sample space are equally likely. It is reasonable to assume that Alexis rolls fair number cubes, so all outcomes in the sample space are equally likely. Therefore, the answer is valid and can serve as a reasonable predictor. 7.2.1: Introducing Conditional Probability
53
Guided Practice: Example 1, continued
You can predict the following: If you roll a pair of number cubes a large number of times and consider all the outcomes that have a sum of 6, then 3 about 5 of those outcomes will have both odd numbers. ✔ 7.2.1: Introducing Conditional Probability
54
Bellringer 8/21/14
1.
Define Set
2.
Define Subset
3.
Define Empty Set/Null Set
4.
Define Equal Set
• • •
Guided Practice Example 2
• • • Of the personal watercraft customers, 75 customers were young old or older). 125 of the 200 customers were age 30 or younger. 50 of these customers rented bicycles, and 75 of them rented personal • Consider the following events that apply to a random customer. • Y: The customer is young (30 years old or younger). • • W: The customer rents a personal watercraft. Are Y and W independent? Compare interpret the results.
( ) ( )
Guided Practice: Example 2, continued 1.
Determine if Y and W are independent.
First, determine the probabilities of each event. ( ) ( ) =
P Y W
= ( ) ( ) 125 200 100 200 = = = = 75 100 75 125 0.625
0.5
= = 0.75
0.6
Of 200 customers, 125 were young.
Of 200 customers, 100 rented a personal watercraft. Of 100 personal watercraft customers, 75 were young. Of 125 young customers, 75 rented a personal watercraft. 7.2.1: Introducing Conditional Probability
57
Guided Practice: Example 2, continued
Y
and and ( ) ¹ ( ) .
( ) ¹ 7.2.1: Introducing Conditional Probability
58
Guided Practice: Example 2, continued 2.
Compare
( ) ( ) = = 0.75
0.6
0.75 > 0.6; therefore, ( ) > ( ) .
7.2.1: Introducing Conditional Probability
59
Guided Practice: Example 2, continued 3.
Interpret the results.
( ) represents the probability that a customer is young given that the customer rents a personal watercraft. ( ) represents the probability that a customer rents a personal watercraft given that the customer is young. 7.2.1: Introducing Conditional Probability
60
Guided Practice: Example 2, continued
The dependence of the events
Y
and
W
means that a customer’s age affects the probability that the customer rents a personal watercraft; in this case, being young increases that probability because ( ) and
W
> ( ) . The dependence of the events
Y
also means that a customer renting a personal watercraft affects the probability that the customer is young; in this case, renting a personal watercraft increases that probability because ( ) > ( ) .
7.2.1: Introducing Conditional Probability
61
Guided Practice: Example 2, continued
( ) > means that it is more likely that a customer is young given that he or she rents a personal watercraft than it is that a customer rents a personal watercraft given that he or she is young. ✔ 7.2.1: Introducing Conditional Probability
62
Bellringer 8/22/14
1.
Define Union
2.
Define Intersection
3.
Define Complement
4.
Define Element
Bellringer 8/25/14
1.
Draw a Venn Diagram where inside of the universal set you have two circles that overlap representing two sets.
a)
Label Set A, Set B, and the Universal Set
b)
Put dots in the intersection of set A and set B
c) d)
Put zig zags over the union of set A and set B Lightly shade in the Complement of A union B
2.
Write the intersection, union, and complement of A and B using the correct notation.
Bellringer 8/26/14
A = {1, 3, 5} B = {2, 4, 6} C= {2, 4}
1.
How would you describe the events A and B? B and C?
2.
How would the Venn Diagram of sets A and B look compared to how we usually draw them? B and C?
•
REVIEW
• Remember that probability is a number from 0 to 1 inclusive or a percent from 0% to occur. The following diagram shows some sample probabilities.
• When all the outcomes of an experiment are equally likely, the probability of an event E, denoted P(E), is given by ( ) = number of outcomes in
E
number of outcomes in the sample space .
• For situations involving more than one event, the Addition Rule sometimes applies. You will learn about this rule.
• • • • •
Key Concepts
Probability is the likelihood of an event. A probability model is a mathematical model for observable facts or occurrences that are assumed to be random. The facts or occurrences are called outcomes. Actions and processes that produce outcomes are called experiments. When all the outcomes of an experiment are assumed to be equally likely, then the probability model is a uniform probability model. The frequency of an event is the number of times it occurs. The relative frequency of an event is the number of times it occurs divided by the number of times the experiment is performed.
• •
Key Concepts, continued
• • • According to the Addition Rule, if A and B are any two events, then the probability of A or B, denoted P(A or B), is given by the formula P(A or B) = P(A) + P(B) – P(A and B).
• • • • Using set notation, the rule is ( È
B
) = + ( Ç
B
) .
The Addition Rule contains four probabilities: P(A or B), P(A), P(B), and P(A and B); if three of them are known, then the equation can be solved to find the fourth.
If events A and B have no outcomes in common, then they are mutually exclusive events, also known as disjoint events. If A and B are mutually exclusive events, then they cannot both occur, so P(A and B) = 0.
This leads to the following special case of the Addition Rule for mutually exclusive events: ( or
B
) = + ( ) , or ( È
B
) = +
Bellringer 8/27/14
• •
Favorite Book Genre
Non-Fiction Sci-Fi/Fantasy Mystery/Suspense Other
Number that chose the genre
10 15 12 3
Relative frequency Totals
The above table represents the experiment where I asked teachers to choose a favorite book genre.
Complete the table by finding the relative frequencies for each answer
Pop Quiz!!
A B 1.
Find 𝐴 ∪ 𝐵 .
2.
Find 𝐴 ∪ 𝐵 .
3.
What elements are in the Universal Set (also known as Sample Space)?
4.
What is the complement of the Universal set?
U
BellRinger 8/29/14
1.
How do you think the day with the sub went?
2.
Did you use your notes when you got stuck on your assignment?
3.
Did you get off task often?
4.
Did the sub show you the answer key?
5.
Did you do everything I asked you to do while the sub was there?
6.
What do you think the note from the sub said?
7.
How do you think I felt when I saw the note from the sub and the work that you turned in? 8.
What could have been done to make yesterday go better?
Bellringer 9/2/14
1.
2.
3.
Draw a Venn Diagram of set A={1, 2, 5, 7, 8} and set B={3, 7, 9, 13} where the Universal set contains numbers 1-15 Do they intersect? How do you know?
Where is the complement of the union of A and B?
• • •
Key Concepts
• • According to the Addition Rule, if A and B are any two events, then the probability of A or B, denoted P(A or B), is given by the formula P(A or B) = P(A) + P(B) – P(A and B).
• • • Using set notation, the rule is ( È
B
) = ( ) + ( ) ( Ç
B
) .
The Addition Rule contains four probabilities: P(A or B), P(A), P(B), and P(A and B); if three of them are known, then the equation can be solved to find the fourth.
If events A and B have no outcomes in common, then they are mutually exclusive events, also known as disjoint events. If A and B are mutually exclusive events, then they cannot both occur, so P(A and B) = 0.
This leads to the following special case of the Addition Rule for mutually exclusive events: ( or
B
) = ( ) + ( ) , or
P A
È
B
) = ( ) + ( )
Bellringer 9/3/14
1.
What are Mutually Exclusive Events?
2.
What are 2 examples of Disjoint Events?
3.
What is the P(A and B) when A and B are Mutually Exclusive?
Example 1
• Donte is playing a dice game with a 20-sided die. He’s hoping for an even number or number greater then 15 on his first roll. What is the probability that he rolls an even number or a number greater then 15 on his first roll?
Guided Practice: Example 1, continued 1.
Identify the sample space and count the outcomes.
The sample space is the set of all cards in the deck, so there are 52 outcomes.
7.1.2: The Addition Rule
76
Guided Practice: Example 1, continued 2.
Identify the outcomes in the event and count the outcomes.
You can use a table to show the sample space. Then identify and count the cards that are either a club or a face card or both a club and a face card.
Suit
Spade Club
2 3 4 5 6 7 8 9 1 0 J Q K A
✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Diamon ✔ ✔ ✔ The event of “club or face card” has 22 outcomes.
Heart ✔ ✔ ✔ 7.1.2: The Addition Rule
77
Guided Practice: Example 1, continued 3.
Apply the formula for the probability of an event.
= number of outcomes in
E
number of outcomes in the sample space
P
( club or face card ) = 22 52 = 11 » 26 0.42
7.1.2: The Addition Rule
78
Guided Practice: Example 1, continued 4.
Apply the Addition Rule to verify your answer.
Let
A
be the event “club.” There are 13 clubs, so
A
has 13 outcomes. Let
B
be the event “face card.” There are 12 face cards, so
B
has 12 outcomes. The event “
A
and
B
” is the event “club and face card,” which has 3 outcomes: jack of clubs, queen of clubs, and king of clubs.
7.1.2: The Addition Rule
79
Guided Practice: Example 1, continued
Apply the Addition Rule.
P
( ( or
B
) = ( ) + club or face card ) =
P
( )
P
( ( ( ) + and
P
(
B
) face card club and face card ) ) -
P
( club or face card ) = 13 52 + 12 52 3 52 = 22 52 = 11 » 0.42
26 7.1.2: The Addition Rule
80
Guided Practice: Example 1, continued
The Addition Rule answer checks out with the probability found in step 3, so the probability of the event is approximately 0.42.
✔ 7.1.2: The Addition Rule
81
Example 2
• Students at Rolling Hills High School receive an achievement award for either performing community service or making the honor roll. The school has 500 students and 180 of them received the award. There were 125 students who performed community service and 75 students who made the honor roll. What is the probability that a randomly chosen student at Rolling Hills High School performed community service and made the honor roll?
Guided Practice: Example 2, continued 1.
Define the sample space and state its number of outcomes.
The sample space is the set of all students at the school; it has 500 outcomes.
7.1.2: The Addition Rule
83
Guided Practice: Example 2, continued 2.
Define events that are associated with the numbers in the problem and state their probabilities.
Let
A
Then be the event “performed community service.” ( ) = 125 500 .
Let
B
be the event “made the honor roll.” Then ( ) = 75 500 .
7.1.2: The Addition Rule
84
Guided Practice: Example 2, continued
The event “
A
or
B
” is the event “performed community service or made the honor roll,” and can also be written
A
È
B
. ( È
B
) = 180 500 because 180 students received the award for either community service or making the honor roll. 7.1.2: The Addition Rule
85
Guided Practice: Example 2, continued 3.
Write the Addition Rule and solve it for P(A and B), which is the probability of the event “performed community service and made the honor roll.” P(A and B) can also be written
7.1.2: The Addition Rule
86
Guided Practice: Example 2, continued
( È
B
) = + ( Ç
B
) 180 500 = 125 500 + 75 500 180 = 500 20 500 200 500 = ( Ç ( Ç
B
)
B
) 20 500 = ( Ç
B
) ( Ç
B
) Substitute the known probabilities.
Simplify. Subtract 200 500 from both sides.
Multiply both sides by –1.
7.1.2: The Addition Rule
87
Guided Practice: Example 2, continued
The probability that a randomly chosen student at Rolling Hills High School has performed community service and is on the honor roll is 20 500 = 1 25 , or 4%.
✔ 7.1.2: The Addition Rule
88
Introduction / Key Concepts
• In probability, events are either dependent or independent. • Two events are independent if the occurrence or non-occurrence of one event has no effect on the probability of the other event. If two events are independent, then you can simply multiply their individual probabilities to find the probability that both events will occur.
• If events are dependent, then the outcome of one event affects the outcome of another event. So it is important to know whether or not two events are independent.
Introduction / Key Concepts Continued
• Two events A and B are independent if and only if – P(A and B) = P(A) • P(B). – But sometimes you only know two of these three probabilities and you want to find the third. In such cases, you can’t test for independence, so you might assess the situation or nature of the experiment and then make an assumption about whether or not the events are independent. Then, based on your assumption, you can find the third probability in the equation.
Guided Practice
• Trevor tosses a coin 3 times. Consider the following events.
A
: The first toss is heads.
B
: The second toss is heads.
•
C
: There are exactly 2 consecutive heads.
For each of the following pairs of events, determine if the events are independent.
A
and
B
(This is
A
Ç
B
in set notation.)
A
and
C
(This is
A
Ç
C
in set notation.)
B
and
C
(This is
B
Ç
C
in set notation.)
Guided Practice: Example 1, continued 1.
List the sample space.
Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 7.1.3: Understanding Independent Events
92
Guided Practice: Example 1, continued 2.
Use the sample space to determine the relevant probabilities.
= 4 8 = 1 2 = 4 8 = 1 2 ( ) = 2 = 8 4 1 There are 4 outcomes with heads first. There are 4 outcomes with heads second.
There are 2 outcomes with exactly 2 consecutive heads. 7.1.3: Understanding Independent Events
93
Guided Practice: Example 1, continued
( ( Ç
B
) = 2 = 8 ( Ç
C
) = 1 8 4 Ç
C
) = 2 = 8 4 1 1 There are 2 outcomes with heads first and heads second. There is 1 outcome with heads first and exactly 2 consecutive heads. There are 2 outcomes with heads second and exactly 2 consecutive heads.
7.1.3: Understanding Independent Events
94
Guided Practice: Example 1, continued 3.
Use the definition of independence to determine if the events are independent in each specified pair.
( Ç
B
) 1 4 = 1 · 2 = 1 2 ( ) · ( )
A
and
B
are independent.
7.1.3: Understanding Independent Events
95
Guided Practice: Example 1, continued
( Ç
C
) = ( ) · 1 8 = 1 · 2 1 4
A
and
C
are independent.
4 1 Ç
C
) ¹ 1 · 2 = 1 4 ( ) · ( )
B
and
C
are dependent.
7.1.3: Understanding Independent Events ✔
96
1.
2.
3.
4.
Bellringer 9/8/14
• Set A = {1,3,5} • Set B = {2,4,6}
Find:
A ∪ B 𝐴 ∩ 𝐵 𝐴 𝐴 ∩ 𝐵
Objectives
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.
Vocabulary
joint relative frequency marginal relative frequency conditional relative frequency
A
two-way table
is a useful way to organize data that can be categorized by two variables. Suppose you asked 20 children and adults whether they liked broccoli. The table shows one way to arrange the data.
The
joint relative frequencies
are the values in each category divided by the total number of values, shown by the shaded cells in the table. Each value is divided by 20, the total number of individuals.
The
marginal relative frequencies
adding the joint relative frequencies in each row and column.
are found by
To find a
conditional relative frequency
, divide the joint relative frequency by the marginal relative frequency. Conditional relative frequencies can be used to find conditional probabilities.
Example 1: Finding Joint and Marginal Relative Frequencies The table shows the results of randomly selected car insurance quotes for 125 cars made by an insurance company in one week. Make a table of the joint and marginal relative frequencies.
Example 1: Continued
Divide each value by the total of 125 to find the joint relative frequencies, and add each row and column to find the marginal relative frequencies.
0 acc.
1 acc.
2 + acc.
Total Teen 0.12
0.032
0.072
0.224
Adult 0.424
0.256
0.096
0.776
Total 0.544
0.288
0.168
1
Bell Ringer 9/9/14
A bag contains 4 red and 2 yellow marbles. A marble is selected, kept out of the bag, and another marble is selected. Find each conditional probability of selecting the second marble.
1.
P
(red | red)
0.6
2.
P
(red | yellow)
0.8
3.
P
(yellow | yellow)
0.2
4.
P
(yellow | red)
0.4
Check It Out!
Example 1 The table shows the number of books sold at a library sale. Make a table of the joint and marginal relative frequencies.
Check It Out!
Example 1 Continued
Divide each value by the total of 210 to find the joint relative frequencies, and add each row and column to find the marginal relative frequencies.
Hardcover Paperback Total Fiction 0.133
0.448
0.581
Nonfiction 0.248
0.171
0.419
Total 0.381
0.619
1
Example 2: Using Conditional Relative Frequency to Find Probability A reporter asked 150 voters if they plan to vote in favor of a new library and a new arena. The table shows the results.
Example 2A Continued A.
Make a table of the joint and marginal relative frequencies.
Arena Yes No Total Library Yes 0.14
0.38
0.52
No 0.2
0.28
0.48
Total 0.34
0.66
1
Example 2B Continued B.
If you are given that a voter plans to vote
no
to the new library, what is the probability the voter also plans to say no to the new arena?
Check It Out!
Example 2 The classes at a dance academy include ballet and tap dancing. Enrollment in these classes is shown in the table.
2a.
Copy and complete the table of the joint relative frequencies and marginal relative frequencies.
Tap
Check It Out!
Example 2 continued
Yes No Total Ballet Yes 0.19
0.43
0.62
No 0.26
0.12
0.38
Total 0.45
0.55
1
2b.
what is the probability that the student is not taking tap?
If you are given that a student is taking ballet, 0.43
0.62
≈ 0.69 or 69%
Bellringer 9/10/14
Kira likes number puzzles and is tackling the puzzle below. The puzzle is incomplete. When it is complete, each total at the end of a row will be the sum of the numbers in that row, and each total at the end of a column will be the sum of the numbers in that column. Find all the missing numbers to solve the puzzle.
Bellringer 9/10/14
Kira likes number puzzles and is tackling the puzzle below. The puzzle is incomplete. When it is complete, each total at the end of a row will be the sum of the numbers in that row, and each total at the end of a column will be the sum of the numbers in that column. Find all the missing numbers to solve the puzzle.
This sum is 25.
This sum is 30.
Totals Totals
10 10 25 40 12 8 30 34 100
Completed puzzle:
Totals
10
5
10 25
14 17 9
40 12 8
15 35 Totals 36
30 34 100 7.2.2: Using Two-Way Frequency Tables
115
Explanation: • • • For the first column, the sum given is 25 and the numbers provided add to 20. Therefore, the missing cell is 5.
The second row has a sum of 30. Using the information completed from the first column, we know that the first cell in the first row is 5. We are given 8 and the total is 30. Therefore, the missing cell is 30 – (8 + 5) = 17.
The total row at the bottom sums to 100 and the first two cells are given as 25 and 40. Use 100 – (25 + 40) = 35 to find the missing number.
7.2.2: Using Two-Way Frequency Tables
116
• • • • The third column we now know sums to 35, and 12 and 8 were given as two entries in that column. Use 35 – (12 + 8) = 15 to find the missing cell.
The third row now has two entries, 10 and 15. The row must sum to 34. Use 34 – (10 + 15) = 9 to find the missing cell.
The second column sums to 40 and we now have two entries in the column, 17 and 9. Use 40 – (17 + 9) = 14 to find the missing cell.
The sum is missing from the first row. All the cells have been filled in for the first row. Sum the cells to find the missing entry: 10 + 14 + 12 = 36.
7.2.2: Using Two-Way Frequency Tables
117
Example 3: Comparing Conditional Probabilities A company sells items in a store, online, and through a catalog. A manager recorded whether or not the 50 sales made one day were paid for with a gift card.
Use conditional probabilities to determine for which method a customer is most likely to pay with a gift card.
Store Online Catalog TOTAL Example 3 Continued Gift Card Another 0.12
0.18
0.10
0.40
Method 0.18
0.26
0.16
0.60
TOTAL 0.30
0.44
0.26
1
P
(gift card if in store) = 0.4
P
(gift card if online) ≈ 0.41
P
(gift card if by catalog) ≈ 0.38
so most likely if buying online.
A customer is most likely to pay with a gift card if buying online.
Check It Out!
Example 3 Francine is evaluating three driving schools. She asked 50 people who attended the schools whether they passed their driving tests on the first try. Use conditional probabilities to determine which is the best school.
Use conditional probabilities to determine which is the best school.
Check It Out!
Example 3 Continued Pass Al’s Driving 0.28
Drive Time Crash Course TOTAL 0.22
0.10
0.60
Fail 0.16
0.14
0.10
0.40
TOTAL 0.44
0.36
0.20
1
Al’s Driving has the best pass rate, about 64%, versus 61% for Drive Time and 50% for Crash Course.
Lesson Quiz: Part I
1. At a juice-bottling factory, quality-control technicians randomly select bottles and mark them
pass
results.
or
fail
. The manager randomly selects the results of 50 tests and organizes the data by shift and result. The table below shows these
Lesson Quiz: Part I continued 1. Make a table of the joint and marginal relative frequencies.
Lesson Quiz: Part 2 2. Find the probability that a bottle was inspected in the afternoon given that it failed the inspection.
Lesson Quiz: Part 3 3. Use conditional probabilities to determine on which shift a bottle is most likely to pass inspection.
Bellringer 9/11/14
Define and Give an Example: 1.
Set 2.
3.
4.
Union Intersection Complement
Bellringer 8/12/14
1.
2.
3.
–
You only have 5 minutes from the time the bell rings!
What is the probability of rolling a 3, and then a 7 on a 6 sided die?
What is the Probability of spinning a spinner with the numbers 1-10 on it and it landing on a 2 or an 8? What is the Probability of spinning a spinner with the numbers 1-10 on it and it landing on a 2 and then an 8?
– Turn in your bellringer and pick up a whiteboard, marker, and eraser when you are finished • No, 2 and 3 are not the same question or answer • Yes we will go over it