Chapter 5 Probability Section 5.2 Probability Rules Probability Rules LEARNING TARGETS By the end of this section, you should be able to: 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. Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. A sample space can be very simple or very complex. Starnes/Tabor, The Practice of Statistics Probability Models Starnes/Tabor, The Practice of Statistics Probability Models Sample Space 36 Outcomes Starnes/Tabor, The Practice of Statistics Probability Models Sample Space 36 Outcomes Since the dice are fair, each outcome is equally likely. Each outcome has probability 1/36. Starnes/Tabor, The Practice of Statistics Probability Models An event is any collection of outcomes from some chance process. Starnes/Tabor, The Practice of Statistics 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 Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. Since each outcome has 1 4 probability , P(A) = . 36 36 Starnes/Tabor, The Practice of Statistics Probability Models Finding Probabilities: Equally Likely Outcomes Starnes/Tabor, The Practice of Statistics 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 Starnes/Tabor, The Practice of Statistics 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 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 2. 3. 4. 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. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 2. The probability of any event is a number between 0 and 1. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 2. 3. The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities that add up to 1. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 2. 3. 4. 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. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Rules that a valid probability model must obey: 1. If all outcomes in the sample space are equally likely, the probability 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝐴 that event A occurs is 𝑃 𝐴 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 2. 3. 4. 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. Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? 5 P(sum is 6) = 36 Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? 5 P(sum is 6) = 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? 5 P(sum is 6) = 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? 4 5 9 P(sum is 5 or sum is 6) = P(sum is 5) + P(sum is 6)= + = = 0.25 36 36 36 Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? 5 P(sum is 6) = 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? 4 5 9 P(sum is 5 or sum is 6) = P(sum is 5) + P(sum is 6)= + = = 0.25 36 36 36 Two events A and B are mutually exclusive (disjoint) 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 (disjoint) events A and B says that P(A or B) = P(A) + P(B) Starnes/Tabor, The Practice of Statistics Basic Probability Rules If we roll two fair dice, what is the probability that the sum is 6? 5 P(sum is 6) = 36 If we roll two fair dice, what is the probability that the sum is 5 or 6? 4 5 9 P(sum is 5 or sum is 6) = P(sum is 5) + P(sum is 6)= + = = 0.25 36 36 36 Two events A and B are mutually exclusive (disjoint) 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 (disjoint) events A and B says that P(A or B) = P(A) + P(B) Starnes/Tabor, The Practice of Statistics Basic Probability Rules We can summarize the basic probability rules in symbolic form. Basic Probability Rules Starnes/Tabor, The Practice of Statistics Basic Probability Rules We can summarize the basic probability rules in symbolic form. Basic Probability Rules • For any event A, 0 ≤ 𝑃(𝐴) ≤ 1. Starnes/Tabor, The Practice of Statistics Basic Probability Rules 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 Starnes/Tabor, The Practice of Statistics Basic Probability Rules 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 Starnes/Tabor, The Practice of Statistics Basic Probability Rules 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 𝐴𝐶 = 1 − 𝑃(𝐴) Starnes/Tabor, The Practice of Statistics Basic Probability Rules 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 𝐴𝐶 = 1 − 𝑃(𝐴) • Addition rule for mutually exclusive events: If A and B are mutually exclusive, 𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵) Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. (a) The probability of each outcome is a number between 0 and 1. The sum of the probabilities is 0.207 + 0.205 + 0.198 + 0.135 + 0.131 + 0.124 = 1. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. (b) Find the probability that you don’t get a blue M&M. Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. (b) Find the probability that you don’t get a blue M&M. (b) P (not blue) = 1 – P(blue) = 1 – 0.207 = 0.793 Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. (b) Find the probability that you don’t get a blue M&M. (c) What’s the probability that you get an orange or a brown M&M? Starnes/Tabor, The Practice of Statistics Basic Probability Rules Problem: Suppose you tear open the corner of a bag of M&M’S® Milk Chocolate Niels Poulsen std/Alamy 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: (a) Explain why this is a valid probability model. (b) Find the probability that you don’t get a blue M&M. (c) What’s the probability that you get an orange or a brown M&M? (c) P (orange or brown) = P (orange) + P (brown) = 0.205 + 0.124 = 0.329 Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule There are two different uses of the word or in everyday life. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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). Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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). (a) P (B) = P (pierced ear) = 103 178 = 0.579 Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. (b) P (A and B) = P (male and pierced ear) = 19 178 = 0.107 Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. (b) There’s about an 11% chance that a randomly selected student from this class is male and has a pierced ear. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. (c) Find P(A or B). Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. (c) Find P(A or B). (c) P (A or B) = P (male or pierced ear) = 71+19+84 178 = 0.978 Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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. Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule We can fix the doublecounting problem illustrated in the twoway table by subtracting the probability P(male and pierced ear) from the sum of P(A) and P(B). Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule We can fix the doublecounting problem illustrated in the twoway 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) 90 103 19 + − 178 178 178 174 = 178 = Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule We can fix the doublecounting problem illustrated in the twoway 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) 90 103 19 + − 178 178 178 174 = 178 = 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) Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule We can fix the doublecounting problem illustrated in the twoway 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) 90 103 19 + − 178 178 178 174 = 178 = The probability P(A and B) that events A and B both occur is called a joint probability. If the joint probability is 0, the events are disjoint. 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) Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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? Starnes/Tabor, The Practice of Statistics Two-Way Tables, Probability, and the General Addition Rule 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? Let event F = uses Facebook and = uses Instagram. P(F or I) = P(F) + P(I) – P(F and I) = 0.68 + 0.28 – 0.25 = 0.71 Ann Heath I Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics 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. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability The complement AC contains exactly the outcomes that are not in A. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. (a) Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. (a) Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. (a) Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. (a) Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. (b) Find the probability that the person uses neither Facebook nor Instagram. Starnes/Tabor, The Practice of Statistics Venn Diagrams and Probability 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. (a) Make a Venn diagram to display the sample space of this chance process using the events F: uses Facebook and I: uses Instagram. (b) Find the probability that the person uses neither Facebook nor Instagram. (b) P(no Facebook and no Instagram) = 0.29 Starnes/Tabor, The Practice of Statistics Section Summary LEARNING TARGETS After this section, you should be able to: 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. Starnes/Tabor, The Practice of Statistics