Pull 2 samples of 3 pennies and record both averages (2 dots).

Pull 2 samples of 3 pennies and record both averages (2 dots).
We are going to discuss 5.2 today.

Schedule for the rest of 2018!

Chapter 5 Probability: What Are the Chances? Section 5.2
Probability Rules

Probability Rules GIVE a probability model for a chance process with equally likely outcomes and USE it to find the probability of an event. USE basic probability rules, including the complement rule and the addition rule for mutually exclusive events. USE a two-way table or Venn diagram to model a chance process and calculate probabilities involving two events. APPLY the general addition rule to calculate probabilities.

Probability Models In Section 5.1, we used simulation to imitate chance behavior. Fortunately, we don’t have to always rely on simulations to determine the probability of a particular outcome. Steve Gorton/Getty Images

Probability Models In Section 5.1, we used simulation to imitate chance behavior. Fortunately, we don’t have to always rely on simulations to determine the probability of a particular outcome. A probability model is a description of some chance process that consists of two parts: a list of all possible outcomes and the probability for each outcome. The list of all possible outcomes is called the sample space. Steve Gorton/Getty Images

Probability Models In Section 5.1, we used simulation to imitate chance behavior. Fortunately, we don’t have to always rely on simulations to determine the probability of a particular outcome. A probability model is a description of some chance process that consists of two parts: a list of all possible outcomes and the probability for each outcome. The list of all possible outcomes is called the sample space. Steve Gorton/Getty Images A sample space can be very simple or very complex.

Probability Models

Probability Models Sample Space 36 Outcomes

Probability Models Sample Space 36 Outcomes Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36.

Probability Models An event is any collection of outcomes from some chance process. Steve Gorton/Getty Images

Probability Models An event is any collection of outcomes from some chance process. Let A = getting a sum of 5 when two fair dice are rolled Steve Gorton/Getty Images

Probability Models An event is any collection of outcomes from some chance process. Let A = getting a sum of 5 when two fair dice are rolled There are 4 outcomes that result in a sum of 5. Steve Gorton/Getty Images

Probability Models An event is any collection of outcomes from some chance process. Let A = getting a sum of 5 when two fair dice are rolled There are 4 outcomes that result in a sum of 5. Steve Gorton/Getty Images Since each outcome has probability , P(A) =

Finding Probabilities: Equally Likely Outcomes
Probability Models Finding Probabilities: Equally Likely Outcomes Steve Gorton/Getty Images

Probability Models Finding Probabilities: Equally Likely Outcomes If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula Steve Gorton/Getty Images

Probability Models Finding Probabilities: Equally Likely Outcomes If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Steve Gorton/Getty Images

Basic Probability Rules
Rules that a valid probability model must obey:

Rules that a valid probability model must obey: If all outcomes in the sample space are equally likely, the probability that event A occurs is 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒

Rules that a valid probability model must obey: If all outcomes in the sample space are equally likely, the probability that event A occurs is 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 The probability of any event is a number between 0 and 1.

Rules that a valid probability model must obey: If all outcomes in the sample space are equally likely, the probability that event A occurs is 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities that add up to 1.

Rules that a valid probability model must obey: If all outcomes in the sample space are equally likely, the probability that event A occurs is 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities that add up to 1. The probability that an event does not occur is 1 minus the probability that the event does occur.

Rules that a valid probability model must obey: If all outcomes in the sample space are equally likely, the probability that event A occurs is 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities that add up to 1. The probability that an event does not occur is 1 minus the probability that the event does occur. The complement rule says that P (AC) = 1 – P(A), where AC is the complement of event A; that is, the event that A does not occur.

If we roll two fair dice, what is the probability that the sum is 6?

If we roll two fair dice, what is the probability that the sum is 6? P(sum is 6) = 5 36

If we roll two fair dice, what is the probability that the sum is 6? P(sum is 6) = 5 36 If we roll two fair dice, what is the probability that the sum is 5 or 6?

If we roll two fair dice, what is the probability that the sum is 6? P(sum is 6) = 5 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? P(sum is 5 or sum is 6)=P(sum is 5)+P(sum is 6)= = 9 36 =0.25

If we roll two fair dice, what is the probability that the sum is 6? P(sum is 6) = 5 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? P(sum is 5 or sum is 6)=P(sum is 5)+P(sum is 6)= = 9 36 =0.25 Two events A and B are mutually exclusive if they have no outcomes in common and so can never occur together—that is, if P(A and B) = 0. The addition rule for mutually exclusive events A and B says that P(A or B) 5= P(A) + P(B)

If we roll two fair dice, what is the probability that the sum is 6? P(sum is 6) = 5 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? P(sum is 5 or sum is 6)=P(sum is 5)+P(sum is 6)= = 9 36 =0.25 Note that this rule works only for mutually exclusive events. CAUTION: Two events A and B are mutually exclusive if they have no outcomes in common and so can never occur together—that is, if P(A and B) = 0. The addition rule for mutually exclusive events A and B says that P(A or B) 5= P(A) + P(B)

We can summarize the basic probability rules in symbolic form. Basic Probability Rules

We can summarize the basic probability rules in symbolic form. Basic Probability Rules For any event A, 0≤𝑃(𝐴)≤1.

We can summarize the basic probability rules in symbolic form. Basic Probability Rules For any event A, 0≤𝑃(𝐴)≤1. If S is the sample space in a probability model, 𝑃 𝑆 =1

We can summarize the basic probability rules in symbolic form. Basic Probability Rules For any event A, 0≤𝑃(𝐴)≤1. If S is the sample space in a probability model, 𝑃 𝑆 =1 In the case of equally likely outcomes, 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒

We can summarize the basic probability rules in symbolic form. Basic Probability Rules For any event A, 0≤𝑃(𝐴)≤1. If S is the sample space in a probability model, 𝑃 𝑆 =1 In the case of equally likely outcomes, 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Complement rule: 𝑃 𝐴 𝐶 =1−𝑃(𝐴)

We can summarize the basic probability rules in symbolic form. Basic Probability Rules For any event A, 0≤𝑃(𝐴)≤1. If S is the sample space in a probability model, 𝑃 𝑆 =1 In the case of equally likely outcomes, 𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Complement rule: 𝑃 𝐴 𝐶 =1−𝑃(𝐴) Addition rule for mutually exclusive events: If A and B are mutually exclusive, 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃(𝐵)

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Niels Poulsen std/Alamy

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Niels Poulsen std/Alamy The probability of each outcome is a number between 0 and 1. The sum of the probabilities is = 1.

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Find the probability that you don’t get a blue M&M. Niels Poulsen std/Alamy

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Find the probability that you don’t get a blue M&M. Niels Poulsen std/Alamy (b) P (not blue) = 1 – P(blue) = 1 – = 0.793

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Find the probability that you don’t get a blue M&M. What’s the probability that you get an orange or a brown M&M? Niels Poulsen std/Alamy

Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Candies, pour one candy into your hand, and observe the color. According to Mars, Inc., the maker of M&M’S, the probability model for a bag from its Cleveland factory is: Explain why this is a valid probability model. Find the probability that you don’t get a blue M&M. What’s the probability that you get an orange or a brown M&M? Niels Poulsen std/Alamy (c) P (orange or brown) = P (orange) + P (brown) = = 0.329

Two-Way Tables, Probability, and the General Addition Rule
There are two different uses of the word or in everyday life.

There are two different uses of the word or in everyday life. When you are asked if you want “soup or salad,” the waiter wants you to choose one or the other, but not both.

There are two different uses of the word or in everyday life. When you are asked if you want “soup or salad,” the waiter wants you to choose one or the other, but not both. However, when you order coffee and are asked if you want “cream or sugar,” it’s OK to ask for one or the other or both.

There are two different uses of the word or in everyday life. When you are asked if you want “soup or salad,” the waiter wants you to choose one or the other, but not both. However, when you order coffee and are asked if you want “cream or sugar,” it’s OK to ask for one or the other or both. In mathematics and probability, “A or B” means one or the other or both.

There are two different uses of the word or in everyday life. When you are asked if you want “soup or salad,” the waiter wants you to choose one or the other, but not both. However, when you order coffee and are asked if you want “cream or sugar,” it’s OK to ask for one or the other or both. In mathematics and probability, “A or B” means one or the other or both. When you’re trying to find probabilities involving two events, like P(A or B), a two-way table can display the sample space in a way that makes probability calculations easier.

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. Find P(B). Big Cheese Photo/Superstock

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. Find P(B). Big Cheese Photo/Superstock (a) P (B) = P (pierced ear) = = 0.579

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. (a) Find P(B). (b) Find P(A and B). Interpret this value in context. Big Cheese Photo/Superstock

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. (a) Find P(B). (b) Find P(A and B). Interpret this value in context. Big Cheese Photo/Superstock (b) P (A and B) = P (male and pierced ear) = = 0.107

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. (a) Find P(B). Find P(A and B). Interpret this value in context. Big Cheese Photo/Superstock (b) There’s about an 11% chance that a randomly selected student from this class is male and has a pierced ear.

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. (a) Find P(B). Find P(A and B). Interpret this value in context. Find P(A or B). Big Cheese Photo/Superstock

Problem: Students in a college statistics class wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables— gender and whether or not the student had a pierced ear—for all 178 people in the class. The two-way table summarizes the data. Suppose we choose a student from the class at random. Define event A as getting a male student and event B as getting a student with a pierced ear. (a) Find P(B). Find P(A and B). Interpret this value in context. Find P(A or B). Big Cheese Photo/Superstock (c) P (A or B) = P (male or pierced ear) = = 0.978

Part (c) of the example reveals an important fact about finding the probability P(A or B): we can’t use the addition rule for mutually exclusive events unless events A and B have no outcomes in common.

We can fix the double-counting problem illustrated in the two-way table by subtracting the probability P(male and pierced ear) from the sum of P(A) and P(B).

We can fix the double-counting problem illustrated in the two-way table by subtracting the probability P(male and pierced ear) from the sum of P(A) and P(B). 𝑃 𝑚𝑎𝑙𝑒 𝑜𝑟 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟 =𝑃 𝑚𝑎𝑙𝑒 +𝑃 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟 −𝑃(𝑚𝑎𝑙𝑒 𝑎𝑛𝑑 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟) = − =

We can fix the double-counting problem illustrated in the two-way table by subtracting the probability P(male and pierced ear) from the sum of P(A) and P(B). 𝑃 𝑚𝑎𝑙𝑒 𝑜𝑟 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟 =𝑃 𝑚𝑎𝑙𝑒 +𝑃 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟 −𝑃(𝑚𝑎𝑙𝑒 𝑎𝑛𝑑 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟) = − = If A and B are any two events resulting from some chance process, the general addition rule says that 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃(𝐴 𝑎𝑛𝑑 𝐵)

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. What’s the probability that the person uses Facebook or uses Instagram? Ann Heath

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. What’s the probability that the person uses Facebook or uses Instagram? Ann Heath Let event F = uses Facebook and I = uses Instagram. P(F or I) = P(F) + P(I) – P(F and I) = – 0.25 = 0.71

Venn Diagrams and Probability
A Venn diagram consists of one or more circles surrounded by a rectangle. Each circle represents an event. The region inside the rectangle represents the sample space of the chance process.

The complement AC contains exactly the outcomes that are not in A.

The complement AC contains exactly the outcomes that are not in A. The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B.

The complement AC contains exactly the outcomes that are not in A. The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B. The union of events A and B (A ∪ B) is the set of all outcomes in either event A or B.

The complement AC contains exactly the outcomes that are not in A. The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B. HINT: To keep the symbols straight, remember ∪ for Union and ∩ for intersection. The union of events A and B (A ∪ B) is the set of all outcomes in either event A or B.

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. Ann Heath

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. Ann Heath (a)

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. Find the probability that the person uses neither Facebook nor Instagram. Ann Heath

Problem: A survey of all residents in a large apartment complex reveals that 68% use Facebook, 28% use Instagram, and 25% do both. Suppose we select a resident at random. Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. Find the probability that the person uses neither Facebook nor Instagram. Ann Heath (b) P(no Facebook and no Instagram) = 0.29

Section Summary GIVE a probability model for a chance process with equally likely outcomes and USE it to find the probability of an event. USE basic probability rules, including the complement rule and the addition rule for mutually exclusive events. USE a two-way table or Venn diagram to model a chance process and calculate probabilities involving two events. APPLY the general addition rule to calculate probabilities.

Assignment 5.2 p #31, 35, 39, 41, 43, 45,47, 51, 53, 55, 57 Self-Check, text me questions, or come to tutoring. If you are stuck on any of these, look at the odd before or after and the answer in the back of your book. If you are still not sure text a friend or me for help (before 8pm). Tomorrow I am out. You review together for 5.2 Quiz on Fri.

Pull 2 samples of 3 pennies and record both averages (2 dots).

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Pull 2 samples of 3 pennies and record both averages (2 dots).

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