# Ch 5: Common Core HS Statistics & Probability: Probability

### About This Chapter

**Standard:** 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'). (CCSS.Math.Content.HSS-CP.A.1)

**Standard:** 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. (CCSS.Math.Content.HSS-CP.A.2)

**Standard:** 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. (CCSS.Math.Content.HSS-CP.A.3)

**Standard:** 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. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. (CCSS.Math.Content.HSS-CP.A.4)

**Standard:** Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (CCSS.Math.Content.HSS-CP.A.5)

**Standard:** 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. (CCSS.Math.Content.HSS-CP.B.6)

**Standard:** 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. (CCSS.Math.Content.HSS-CP.B.7)

**Standard:** 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. (CCSS.Math.Content.HSS-CP.B.8)

**Standard:** Use permutations and combinations to compute probabilities of compound events and solve problems. (CCSS.Math.Content.HSS-CP.B.9)

## About This Chapter

A clear understanding of probability enables students to calculate the likelihood of simple, compound, complementary, independent and dependent events. They can use two-way tables to determine whether events are independent and to estimate conditional probabilities for real-life situations.

Our lessons for these standards address concepts such as:

- Elements, intersections and unions in mathematical sets
- Subsets of a sample space
- Probability of simple, compound, complementary, independent and dependent events
- Conditional probabilities
- Addition and multiplication rules and probability
- Probability of permutations

Students who grasp these concepts are able to calculate conditional probabilities, permutations and the probability of permutations. They can prepare for college and careers, where the ability to apply probability to research and real life situations is an asset.

### How to Use These Lessons in Your Classroom

The following are some tips for injecting lessons on probability into your curriculum to meet the common core standards:

#### Set the Purpose for Learning Lessons

Prior to viewing the videos, challenge your students to complete the corresponding quizzes for several of the following lessons: *Mathematical Sets: Elements, Intersections & Unions; Probability of Simple, Compound and Complementary Events; Probability of Independent and Dependent Events; Probability of Independent Events: The 'At Least One' Rule; How to Calculate Simple Conditional Probabilities; How to Calculate a Permutation* or *How to Calculate the Probability of Permutations*. Ask students to re-take the quizzes after completing the video lessons to assess understanding.

#### Addition Rule Lesson

Watch the video lesson pertaining to probability and the addition rule. Using an online, interactive whiteboard or hand-held dies, multi-colored spinners and beads in jars, students will predict the possibilities. They will then roll/spin/choose, using technology or manipulatives, and note the outcomes. Discuss and discover how the addition rule applies to their results.

#### Conditional Probabilities, Independent Events and Multiplication Rule Lessons

Share the pertinent videos from the topics listed above. Using a standard deck of cards, determine the probability that the first card chosen will be a named face card and the second card picked will be another specified face card without the first one being replaced in the deck. Discuss why the events are dependent and why the probability has changed. Guide the students in using the multiplication rule to calculate the probability of events.

### 1. Mathematical Sets: Elements, Intersections & Unions

Today we're going to explore mathematical sets, which are surprisingly simple! Sets are just collections of any objects or concepts, also known as elements, that can be related to each other through union or intersection.

### 2. Events as Subsets of a Sample Space: Definition & Example

Probability can get very confusing at times. You will find that some words, such as events and subsets, are often referring to the same concept depending on the experiment. Use this lesson to understand the concept of events as subsets.

### 3. Probability of Independent and Dependent Events

Sometimes probabilities need to be calculated when more than one event occurs. These types of compound events are called independent and dependent events. Through this lesson, we will look at some real-world examples of how to calculate these probabilities.

### 4. Probability of Independent Events: The 'At Least One' Rule

Occasionally when calculating independent events, it is only important that the event happens once. This is referred to as the 'At Least One' Rule. To calculate this type of problem, we will use the process of complementary events to find the probability of our event occurring at least once.

### 5. How to Calculate Simple Conditional Probabilities

Conditional probability, just like it sounds, is a probability that happens on the condition of a previous event occurring. To calculate conditional probabilities, we must first consider the effects of the previous event on the current event.

### 6. The Relationship Between Conditional Probabilities & Independence

Conditional and independent probabilities are a basic part of learning statistics. It's important that you can understand the similarities and differences between the two as discussed in this lesson.

### 7. Using Two-Way Tables to Evaluate Independence

If you are a visual person, a 2-way table is a great way to analyze information. This lesson shows you how to use a 2-way table to determine the independence of variables.

### 8. Applying Conditional Probability & Independence to Real Life Situations

It can be really confusing learning how to apply conditional and independent probability to real-life situations. This lesson focuses on several examples and practice problems to help you learn how to find conditional probability.

### 9. The Addition Rule of Probability: Definition & Examples

In this lesson, you will learn the differences between mutually exclusive and non-mutually exclusive events and how to find the probabilities of each using the Addition Rule of Probability.

### 10. The Multiplication Rule of Probability: Definition & Examples

The Multiplication Rule of Probability is a concept you will use frequently when solving probability equations. In this lesson, learn the two different scenarios in which you will use the multiplication rule of probability.

### 11. Math Combinations: Formula and Example Problems

Combinations are an arrangement of objects where order does not matter. In this lesson, the coach of the Wildcats basketball team uses combinations to help his team prepare for the upcoming season.

### 12. How to Calculate a Permutation

A permutation is a method used to calculate the total outcomes of a situation where order is important. In this lesson, John will use permutations to help him organize the cards in his poker hand and order a pizza.

### 13. How to Calculate the Probability of Permutations

In this lesson, you will learn how to calculate the probability of a permutation by analyzing a real-world example in which the order of the events does matter. We'll also review what a factorial is. We will then go over some examples for practice.

### 14. Probability of Simple, Compound and Complementary Events

Simple, compound, and complementary events are different types of probabilities. Each of these probabilities are calculated in a slightly different fashion. In this lesson, we will look at some real world examples of these different forms of probability.

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### Other Chapters

Other chapters within the Common Core Math - Statistics & Probability: High School Standards course