College and Career-Readiness Conference Summer 2015 ALGEBRA 2 CONDITIONAL PROBABILITY

TODAY’S OUTCOMES Participants will: 1.Review Coherence as related to Conditional Probability Standards. 2.Take an in-depth look at the S-CP standards taught in Algebra 2. 3.Share best practices and identify muddy points.

Introductions

Cluster A. Understand independence and conditional probability and use them to interpret data Cluster B. Use the rules of probability to compute probabilities of compound events.

OUTCOME 1 Participants will: 1.Review Coherence as related to Conditional Probability Standards.

A purposeful placement of standards to create logical sequences of content topics that bridge across the grades and courses, as well as across standards within each grade/course.

In what grade/subject do students:  Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Construct and interpret a two-way table summarizing data on two categorical variables. - Grade 7 - Grade 8 - Algebra 2

From HSA to PARCC  Students will calculate theoretical probability or use simulations or statistical inference from data to estimate the probability of an event. HSA or PARCC?

HSA - Statistics 3.1.3The student will calculate theoretical probability or use simulations or statistical inference from data to estimate the probability of an event The student will make informed decisions and predictions based upon the results of simulations and data from research.

PARCC Model Content Framework Algebra 2

PARCC Evidence Statements Algebra 2 - EOY S-CP.Int.1Solve multi-step contextual word problems with degree of difficulty appropriate to the course requiring application of course-level knowledge and skills articulated in S-CP i.) Calculating expected values of a random variable is a plus standard not assessed, however, the word “expected” may be used informally (e.g. if you tossed a fair coin 20 times, how many heads would you expect?).

Cluster A. Understand independence and conditional probability and use them to interpret data Standard 1. … unions, intersections, or complements. Standard 2. … independent if the probability of A and B occurring together is the product of their probabilities… Standard 3. Understand the conditional probability of A given B as P(A and B)/P(B)… Standard 4. Construct and interpret two-way frequency tables… Standard 5. Recognize and explain the concepts of conditional probability and independence..

Cluster B. Use the rules of probability to compute probabilities of compound events. Standard 6. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. Standard 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

Unions, Intersections, & Complementary Events – Venn Diagrams AB

Addition Rule of Probability S.CP.7  Formal Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) OR AB

Example  In a certain town, 40% of the people have brown hair, 25 % have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town. Use this given information to answer the following questions. A. If a person is randomly selected, what is the probability that s/he has brown hair and not brown eyes? B. If a person is randomly selected, what is the probability that s/he has brown eyes and not brown hair? C. What is the probability that the person has neither brown hair nor brown eyes?

Create a “Picture” to Model the Problem In a certain town, 40% of the people have brown hair, 25 % have brown eyes, and 15% have both brown hair and brown eyes.

Create a “Picture” to Model the Problem In a certain town, 40% of the people have brown hair, 25 % have brown eyes, and 15% have both brown hair and brown eyes. Let H = people with Brown Hair Let E = people with Brown Eyes HE

Create a “Picture” to Model the Problem In a certain town, 40% of the people have brown hair, 25 % have brown eyes, and 15% have both brown hair and brown eyes. Let H = people with Brown Hair Let E = people with Brown Eyes P(H)=0.40

Solutions

Solution – Two Way Table Brown Hair YesNoTotal Brown Eyes Yes No Total

Essential Skills and Knowledge  S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S-CP.A.1  Ability to describe a sample space.  Understanding of and ability to use set notation, key vocabulary and graphic organizers linked to these standards.

Essential Skills and Knowledge  S-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. S-CP.A.4  Ability to connect experience with two-way frequency tables from Algebra 1 to sample spaces.  Knowledge of characteristics of conditional probability

Essential Skills and Knowledge  S-CP.B.6 – Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S-CP.B.6  Ability to analyze a situation to determine the conditional probability of a described event given that another event occurs.

The Titanic - Task Survived Did not survive Total First class passengers Second class passengers Third class passengers Total passengers Share two thoughts or questions you have about the two – way table.

Making Connections – Two way tables Calculate the following probabilities. Round your answers to three decimal places. 1) If one of the passengers is randomly selected, what is the probability that this passenger was in first class? 2) If one of the passengers is randomly selected, what is the probability that this passenger survived? 3) If one of the passengers is randomly selected, what is the probability that this passenger was in first class and survived?

Solutions

Making Connections – Two way tables Calculate the following probabilities. Round your answers to three decimal places. 4)If one of the passengers is randomly selected from the first class passengers, what is the probability that this passenger survived? (That is, what is the probability that the passenger survived, given that this passenger was in first class?) 5)If one of the passengers who survived is randomly selected, what is the probability that this passenger was in first class? 6)If one of the passengers who survived is randomly selected, what is the probability that this passenger was in third class?

Conditional Probability

Solutions…

PARCC Task – EOY #28 Part A

Independence  Definition – Two events are independent if the occurrence of one event does not effect the probability of the occurrence of the other event.  The following four statements are equivalent  A and B are independent events  P(A and B) = P(A) * P(B)  P(A|B) = P(A) “Probability of A given B”  P(B|A) = P(B) “Probability of B given A”

Independence

Sorting Activity  Sort the cards into two categories. dependent or independent events.  Be ready to discuss.

Solutions Dependent Selecting a king from a standard deck, not replacing it, and then selecting a queen from the same deck. Driving 85 miles per hour, and then getting in a car accident. Smoking a pack of cigarettes per day, and developing emphysema, a chronic lung disease. Returning a rented movie after the due date, and receiving a late fee. A red candy is selected from a package with 30 colored candies and eaten. A blue … Researchers found that people with depression are five times more likely… Independent Tossing a coin and getting a head, and then rolling a six-sided die and obtaining a 6. Exercising frequently and having a 4.0 grade point average Tossing a coin four times and getting four heads, and then tossing it a fifth time and getting heads. The events of getting two aces when two cards are drawn from a deck of playing cards and the first card is replaced before the second card is drawn.

PARCC Task cont…. #28 Part B

Common Core State Standards for Mathematical Practice  1. Make sense of problems and persevere in solving them.  2. Reason abstractly and quantitatively.  4. Model with mathematics.  5. Use appropriate tools strategically.

What have you done that works? Best Practices

Additional Resources  Illustrative Mathematics Illustrative Mathematics  PARCC Practice Test PARCC Practice Test  Engage NY Module Engage NY Module  Mathematics Vision Project Mathematics Vision Project  American Statistical Association American Statistical Association

What are the muddiest points? Record any question you still have after today’s presentation on your post-it note. Please provide your name and address. Stick your post-it on the door as you leave today, and we will respond. Thank you!

Teaching the Common Core content using the Standards for Mathematical Practice to reach progressively higher levels of proficiency attains mathematical rigor. -Hull, Balka, and Harbin Miles