Probability in Common Core Math II Greta Lumsden and Caroline Brooks

Vertical Alignment In the 7 th grade, students are exposed to their first probability unit. The learner objectives for probability in 7 th grade are as follows: As a result of learning, students should be able to… ◦ understand the difference between theoretical and experimental probability ◦ make and utilize a tree diagram ◦ determine payoff of a game using theoretical probability ◦ use an area model to find compound probability ◦ determine expected value ◦ analyze binomial outcomes

Vertical Alignment In CCMI, students are introduced to two-way frequency tables and finding conditional probabilities using two-way frequency tables. The probability-related learner objectives in CCM1 are as follows: By the end of this unit, students will be able to: ◦ Distinguish between categorical and quantitative data ◦ Summarize categorical data by constructing a two-way frequency table ◦ State a hypothesis that can be addressed by the collected data. ◦ Calculate and interpret joint frequencies, marginal frequencies, and conditional frequencies based on a two way table. ◦ Recognize possible associations and trends in categorical data by looking at a two way table.

Probability in CCMII Divided into three major lessons Lesson 1 covers sample spaces, sets and subsets, and basic probability. ◦ S-CP.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”) with visual representations including Venn diagrams. ◦ ◦ S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. ◦ ◦ S-CP.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. ◦ ◦ S-CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. ◦ Instructional Guide Instructional Guide

Probability in CCMII Lesson 2 covers conditional probability and probability using permutations and combinations. ◦ S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. ◦ ◦ S-CP.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.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. ◦ ◦ S-CP.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.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. ◦ ◦ S-CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. ◦ Instructional Guide Instructional Guide

Probability in CCMII Lesson 3 compares theoretical and experimental probability. ◦ S-IC.2 Decide if a specified model is consistent with results from a given data- generating process, e.g., using simulation. ◦ S-IC.6 Evaluate reports based on data. ◦ Instructional Guide Instructional Guide

Lesson 1: Sample Spaces, Subsets, and Basic Probability CCM2 Unit 6: Probability

Sample Space Sample Space: The set of all possible outcomes of an experiment. List the sample space, S, for each of the following: a. Tossing a coin  S = {H,T} b. Rolling a six-sided die  S = {1,2,3,4,5,6} c. Drawing a marble from a bag that contains two red, three blue and one white marble  S = {red, red, blue, blue, blue, white}

Intersections and Unions of Sets The intersection of two sets (A  B) is the set of all elements in both set A AND set B. The union of two sets (A  B) is the set of all elements in set A OR set B. Example: Given the following sets, find A  B and A  B A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15} A  B = {3,9,15} A  B = {0,1,3,5,6,7,9,11,12,13,15}

Venn Diagrams Sometimes drawing a diagram helps in finding intersections and unions of sets. A Venn Diagram is a visual representation of sets and their relationships to each other using overlapping circles. Each circle represents a different set.

Compliment of a set The complement of a set is the set of all elements NOT in the set. ◦ The compliment of a set, A, is denoted as A C Ex: S = {…-3,-2,-1,0,1,2,3,4,…} A = {…-2,0,2,4,…} If A is a subset of S, what is A C ? A C = {-3,-1,1,3,5,…}

Basic Probability Probability of an event occurring is: P(E) = Number of Favorable Outcomes Total Number of Outcomes We can use sample spaces, intersections, unions, and compliments of sets to help us find probabilities of events.  Note that P(A C ) is every outcome except (or not) A, so we can find P(A C ) by finding 1 – P(A)  Why do you think this works?

Odds The odds of an event occurring are equal to the ratio of favorable outcomes to unfavorable outcomes. Odds = Favorable Outcomes Unfavorable Outcomes

In a class of 60 students, 21 sign up for chorus, 29 sign up for band, and 5 take both. 15 students in the class are not enrolled in either band or chorus. 6. Put this information into a Venn Diagram. If the sample space, S, is the set of all students in the class, let students in chorus be set A and students in band be set B. 7. What is A  B? What does it mean in the context of the problem? 8. What is A  B? What does it mean in the context of the problem?

S. Students in the class 15 A. Students in Chorus 5 16 B. Students in Band 24 A  B = 45 A  B = 5

9. What is A C ?B C ? What is (A  B) C ? What is (A  B) C ? 15 What do these mean in the context of the problem? S. Students in the class 15 A. Students in Chorus 16 5 B. Students in Band 24

A card is drawn at random from a standard deck of cards. Find each of the following: 16. P(heart) 13/52 or ¼ 17. P(black card) 26/52 or ½ 18. P(2 or jack) 8/52 or 2/ P(not a heart) 39/52 or 3/4

20. The weather forecast for Saturday says there is a 75% chance of rain. What are the odds that it will rain on Saturday? What does the 75% in this problem mean?  In 100 days where conditions were the same as Saturday, it rained on 75 of those days. The favorable outcome in this problem is that it rains:  75 favorable outcomes, 25 unfavorable outcomes  Odds(rain) = 75/25 or 3/1 Should you make outdoor plans for Saturday?

21. What are the odds of drawing an ace at random from a standard deck of cards? Odds(ace) = 4/48 = 1/12

Probability of Independent and Dependent Events CCM2 Unit 6: Probability

Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability of the other event occurring. Dependent Events: two events are dependent if the outcome or probability of the first event affects the outcome or probability of the second.

Independent Events Suppose a die is rolled and then a coin is tossed. Explain why these events are independent.  They are independent because the outcome of rolling a die does not affect the outcome of tossing a coin, and vice versa. We can construct a table to describe the sample space and probabilities: Roll 1Roll 2Roll 3Roll 4Roll 5Roll 6 Head Tail

How many outcomes are there for rolling the die?  6 outcomes How many outcomes are there for tossing the coin?  2 outcomes How many outcomes are there in the sample space of rolling the die and tossing the coin?  12 outcomes Roll 1Roll 2Roll 3Roll 4Roll 5Roll 6 Head1,H2,H3,H4,H5,H6,H Tail1,T2,T3,T4,T5,T6,T

Is there another way to decide how many outcomes are in the sample space?  Multiply the number of outcomes in each event together to get the total number of outcomes. Let’s see if this works for another situation. Roll 1Roll 2Roll 3Roll 4Roll 5Roll 6 Head1,H2,H3,H4,H5,H6,H Tail1,T2,T3,T4,T5,T6,T

A fast food restaurant offers 5 sandwiches and 3 sides. How many different meals of a sandwich and side can you order? If our theory holds true, how could we find the number of outcomes in the sample space?  5 sandwiches x 3 sides = 15 meals Make a table to see if this is correct. Were we correct? Sand. 1Sand. 2Sand. 3Sand. 4Sand. 5 Side 11,12,13,14,15,1 Side 21,22,23,24,25,2 Side 33,13,23,33,43,5

Probabilities of Independent Events The probability of independent events is the probability of both occurring, denoted by P(A and B) or P(A  B).

Multiplication Rule of Probability The probability of two independent events occurring can be found by the following formula: P(A  B) = P(A) x P(B)

Examples 1. At City High School, 30% of students have part-time jobs and 25% of students are on the honor roll. What is the probability that a student chosen at random has a part-time job and is on the honor roll? Write your answer in context. P(PT job and honor roll) = P(PT job) x P(honor roll) =.30 x.25 =.075 There is a 7.5% probability that a student chosen at random will have a part-time job and be on the honor roll.

2. The following table represents data collected from a grade 12 class in DEF High School. Suppose 1 student was chosen at random from the grade 12 class. (a) What is the probability that the student is female? (b) What is the probability that the student is going to university? Now suppose 2 people both randomly chose 1 student from the grade 12 class. Assume that it's possible for them to choose the same student. (c) What is the probability that the first person chooses a student who is female and the second person chooses a student who is going to university?

Dependent Events Remember, we said earlier that  Dependent Events: two events are dependent if the outcome or probability of the first event affects the outcome or probability of the second. Let’s look at some scenarios and determine whether the events are independent or dependent.

Probabilities of Dependent Events We cannot use the multiplication rule for finding probabilities of dependent events because the one event affects the probability of the other event occurring. Instead, we need to think about how the occurrence of one event will effect the sample space of the second event to determine the probability of the second event occurring. Then we can multiply the new probabilities.

Examples 1. Suppose a card is chosen at random from a deck, the card is NOT replaced, and then a second card is chosen from the same deck. What is the probability that both will be 7s?  This is similar the earlier example, but these events are dependent? How do we know?  How does the first event affect the sample space of the second event?

1. Suppose a card is chosen at random from a deck, the card is NOT replaced, and then a second card is chosen from the same deck. What is the probability that both will be 7s? Let’s break down what is going on in this problem:  We want the probability that the first card is a 7, or P(1 st is 7), and the probability that the second card is a 7, or P(2 nd is 7).  P(1 st is 7) = 4/52 because there a four 7s and 52 cards  How is P(2 nd is 7) changed by the first card being a 7?  P(2 nd is 7) = 3/51  P(1 st is 7, 2 nd is 7) = 4/52 x 3/51 = 1/221 or.0045  The probability of drawing two sevens without replacement is 0.45%

Mutually Exclusive and Inclusive Events CCM2 Unit 6: Probability

Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or you roll a 2?  Can these both occur at the same time? Why or why not? Mutually Exclusive Events (or Disjoint Events): Two or more events that cannot occur at the same time. The probability of two mutually exclusive events occurring at the same time, P(A and B), is 0! Video on Mutually Exclusive Events

Probability of Mutually Exclusive Events To find the probability of one of two mutually exclusive events occurring, use the following formula: P(A or B) = P(A) + P(B) or P(A  B) = P(A) + P(B)

2. Two fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal to 10? Are these events mutually exclusive? Sometimes using a table of outcomes is useful. Complete the following table using the sums of two dice: Die

P(getting a sum less than 7 OR sum of 10) = P(sum less than 7) + P(sum of 10) = 15/36 + 3/36 = 18/36 = ½ The probability of rolling a sum less than 7 or a sum of 10 is ½ or 50%. Die

Mutually Inclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or a number less than 4?  Can these both occur at the same time? If so, when? Mutually Inclusive Events: Two events that can occur at the same time. Video on Mutually Inclusive Events

Probability of the Union of Two Events: The Addition Rule We just saw that the formula for finding the probability of two mutually inclusive events can also be used for mutually exclusive events, so let’s think of it as the formula for finding the probability of the union of two events or the Addition Rule: P(A or B) = P(A  B) = P(A) + P(B) – P(A  B) ***Use this for both Mutually Exclusive and Inclusive events***

Examples 1. What is the probability of choosing a card from a deck of cards that is a club or a ten? P(choosing a club or a ten) = P(club) + P(ten) – P(10 of clubs) = 13/52 + 4/52 – 1/52 = 16/52 = 4/13 or.308 The probability of choosing a club or a ten is 4/13 or 30.8%

3. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the first 10 letters of the alphabet on it or randomly choosing a tile with a vowel on it? P(one of the first 10 letters or vowel) = P(one of the first 10 letters) + P(vowel) – P(first 10 and vowel) = 10/26 + 5/26 – 3/26 = 12/26 or 6/13 The probability of choosing either one of the first 10 letters or a vowel is 6/13 or 46.2%

Lesson 1 in practice Robot Rover

Lesson 2: Conditional Probability CCM2 Unit 6: Probability

Lesson 2 Conditional Probability Lucky Dip Warm-up

Conditional Probability Conditional Probability: A probability where a certain prerequisite condition has already been met. For example:  What is the probability of selecting a queen given an ace has been drawn and not replaced.  What is the probability that a student in the 10 th grade is enrolled in biology given that the student is enrolled in CCM2? Video about Conditional Probability

Conditional Probability Formula The conditional probability of B given A is expressed as P(B | A) P(B | A) = P(B and A) P(A)

Joint Probability P(A and B) S A B

Conditional Probability S P(A and B) represents the outcomes from B that are included in A Since Event A has happened, the sample space is reduced to the outcomes in A A

Examples 1. You are playing a game of cards where the winner is determined by drawing two cards of the same suit. What is the probability of drawing clubs on the second draw if the first card drawn is a club? P(club  club) = P(2 nd club and 1 st club)/P(1 st club) = (13/52 x 12/51)/(13/52) = 12/51 or 4/17 The probability of drawing a club on the second draw given the first card is a club is 4/17 or 23.5%

3. In Mr. Jonas' homeroom, 70% of the students have brown hair, 25% have brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes? P(brown eyes  brown hair) = P(brown eyes and brown hair)/P(brown hair) =.05/.7 =.071 The probability of a student having brown eyes given he or she has brown hair is 7.1%

Using Two-Way Frequency Tables to Compute Conditional Probabilities In CCM1 you learned how to put data in a two-way frequency table (using counts) or a two-way relative frequency table (using percents), and use the tables to find joint and marginal frequencies and conditional probabilities. Let’s look at some examples to review this.

2. The manager of an ice cream shop is curious as to which customers are buying certain flavors of ice cream. He decides to track whether the customer is an adult or a child and whether they order vanilla ice cream or chocolate ice cream. He finds that of his 224 customers in one week that 146 ordered chocolate. He also finds that 52 of his 93 adult customers ordered vanilla. Build a two-way frequency table that tracks the type of customer and type of ice cream. VanillaChocolateTotal Adult Child Total

a. Find P(vanilla  adult) = 52/93 = 55.9% b. Find P(child  chocolate) = 105/146 =71.9% VanillaChocolateTotal Adult5293 Child Total VanillaChocolateTotal Adult Child Total

3. A survey asked students which types of music they listen to? Out of 200 students, 75 indicated pop music and 45 indicated country music with 22 of these students indicating they listened to both. Use a Venn diagram to find the probability that a randomly selected student listens to pop music given that they listen country music. 102 Pop Country 23

P(Pop  Country) = 22/(22+23) = 22/45 or 48.9% 48.9% of students who listen to country also listen to pop. 102 Pop Country 23

Using Conditional Probability to Determine if Events are Independent If two events are statistically independent of each other, then: P(A  B) = P(A) and P(B  A) = P(B) Let’s revisit some previous examples and decide if the events are independent.

13 clubs out of 52 cards Only 12 clubs left and only 51 cards left

13 clubs out of 52 cards Still 13 clubs out of 52 cards

This time we are using the formula for conditional probability

4. Determine whether age and choice of ice cream are independent events. We could start by looking at the P(vanilla  adult) and P(vanilla). If they are the same, then the events are independent. P(vanilla  adult) = 52/93 = 55.9% P(vanilla) = 78/224 = 34.8% P(vanilla  adult)  P(vanilla), so the events are dependent! VanillaChocolateTotal Adult Child Total

Permutations and Combinations

Objectives:  apply fundamental counting principle  compute permutations  compute combinations  distinguish permutations vs combinations  find probabilities using permutations and combinations

Permutations A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

Permutations To find the number of Permutations of n items chosen r at a time, you can use the formula for finding P(n,r) or n P r

Permutations A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? Practice:

Permutations on the Calculator You can use your calculator to find permutations To find the number of permutations of 10 items taken 6 at a time ( 10 P 6 ): Type the total number of items Go to the MATH menu and arrow over to PRB Choose option 2: nPr Type the number of items you want to order

Combinations A Combination is an arrangement of items in which order does not matter. ORDER DOES NOT MATTER! Since the order does not matter in combinations, there are fewer combinations than permutations. The combinations are a "subset" of the permutations.

Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula

Combinations To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? Practice:

Combinations on the Calculator You can use your calculator to find combinations To find the number of combinations of 10 items taken 6 at a time ( 10 C 6 ): Type the total number of items Go to the MATH menu and arrow over to PRB Choose option 3: nCr Type the number of items you want to order

Probability with Permutations and Combinations The 25-member senior class council is selecting officers for president, vice president and secretary. Emily would like to be president, David would like to be vice president, and Jenna would like to be secretary. If the offices are filled at random, beginning with president, what is the probability that they are selected for these offices?

Probability with Permutations and Combinations The 25-member senior class council is selecting members for the prom committee. Stephen, Marcus and Sabrina want would like to be on this committee. If the members are selected at random, what is the probability that all three are selected for this committee?

Lesson 3: Theoretical vs. Experimental Probability Defy the Pig Warm-up