4 Probability Lesson 4 2 Basic Probability Rules

4 Probability Lesson 4. 2 Basic Probability Rules Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Basic Probability Rules Learning Targets After this lesson, you should be able to: ü Give a probability model for a chance process with equally likely outcomes and use it to find the probability of an event. ü Use the complement rule to find probabilities. ü Use the addition rule for mutually exclusive events to find probabilities. Statistics and Probability with Applications, 3 rd Edition 2

Think About It Consider rolling two fair, six-sided dice. Would you be more likely to get a sum of 5 or a sum of 12? Why? Statistics and Probability with Applications, 3 rd Edition 3

Basic Probability Rules Many board games involve rolling dice. Imagine rolling two dice – one that’s red and one that’s blue. We can use the outcomes to develop a probability model. Statistics and Probability with Applications, 3 rd Edition 4

Basic Probability Rules Probability Model, Sample Space 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 probability model does more than just assign a probability to each outcome. It allows us to find the probability of an event. Event An event is any collection of outcomes from some chance process. Statistics and Probability with Applications, 3 rd Edition 5

Basic Probability Rules Events are usually designated by capital letters, like A, B, C, and so on. It is fairly easy to find the probability of an event in the case of equally likely outcomes. 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 Statistics and Probability with Applications, 3 rd Edition 6

Am I blue? Probability models: equally likely outcomes PROBLEM: Suppose you spin a spinner with three equal sections (red, blue, and yellow) two times. (a) Give a probability model for this chance process. The sample space is: RR RB RY BR BB BY YR YB YY Because the spinner has equal sections, each of these 9 outcomes will be equally likely and have probability 1/9. (b) Define event A as spinning blue at least once. Find P(A). There are five outcomes—RB BR BB BY YB—where blue is spun at least once. So Statistics and Probability with Applications, 3 rd Edition 7

Basic Probability Rules Valid probability models must obey basic rules: • The probability of any event is a number between 0 and 1. This rule follows from the definition of probability: the proportion of times the event would occur in many repetitions of the chance process. • All possible outcomes together must have probabilities that add up to 1. Any time we observe a chance process, some outcome must occur. • The probability that an event does not occur is 1 minus the probability that the event does occur. We refer to the event “not A” as the complement of A and denote it by AC. Complement Rule, Complement 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 happen. Statistics and Probability with Applications, 3 rd Edition 8

What’s in the Mystery Box? Complement rule PROBLEM: Mrs. Tyson keeps a Mystery Box in her classroom. If a student meets expectations for behavior, he is allowed to draw a slip of paper without looking. The slips are all of equal size, well mixed, and have the name of a prize written on them. Here is the probability model for the prizes a student can win: Prize Pencil Candy Stickers Homework pass Extra recess time Probability 0. 40 0. 25 0. 15 0. 05 (a) Explain why this is a valid probability model. • The probability of each outcome is a number between 0 and 1 • The probabilities of all the possible outcomes add to 1. Statistics and Probability with Applications, 3 rd Edition 9

What’s in the Mystery Box? Complement rule PROBLEM: Mrs. Tyson keeps a Mystery Box in her classroom. If a student meets expectations for behavior, he is allowed to draw a slip of paper without looking. The slips are all of equal size, well mixed, and have the name of a prize written on them. Here is the probability model for the prizes a student can win: Prize Pencil Candy Stickers Homework pass Extra recess time Probability 0. 40 0. 25 0. 15 0. 05 (b) Find the probability that a student won’t win extra recess time. P(not extra recess) = 1 – P(extra recess) = 1 – 0. 05 = 0. 95. There is a 95% chance a student won’t win extra recess time. Statistics and Probability with Applications, 3 rd Edition 10

Basic Probability Rules When two events have no outcomes in common, we say they are mutually exclusive. Mutually Exclusive, Addition Rule for Mutually Exclusive Events 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) = P(A) + P(B) Note that this rule only works for mutually exclusive events. We will develop a more general rule for finding P(A or B) that works for any two events in Lesson 4. 3. Statistics and Probability with Applications, 3 rd Edition 11

How many people live in U. S. households? Addition rule for mutually exclusive events PROBLEM: According to recent U. S. Census Bureau data (http: //www. census. gov/hhes/families/files/hh 4. xls), the number of people in the household have the following probability model: Number of 7 or 1 2 3 4 5 6 people more Probability 0. 28 0. 34 0. 16 0. 13 0. 06 0. 02 0. 01 Suppose that a household of more than 4 people is considered a large household. Randomly select a U. S. household. (a) Find the probability that the number of people in a household is 4 or less. By the addition rule for mutually exclusive events, P(less than or equal to 4 people) = P(1 or 2 or 3 or 4 people) = P(1 person) + P(2 people) + P(3 people) + P(4 people) = 0. 28 + 0. 34 + 0. 16 + 0. 13 = 0. 91 Statistics and Probability with Applications, 3 rd Edition 12

How many people live in U. S. households? Addition rule for mutually exclusive events Number of people 1 2 3 4 5 6 7 or more Probability 0. 28 0. 34 0. 16 0. 13 0. 06 0. 02 0. 01 (b) Find the probability that the chosen household is a large household. By the complement rule, P(large household) = 1 – P(less than or equal to 4 people) = 1 – 0. 91 = 0. 09. Statistics and Probability with Applications, 3 rd Edition 13

LESSON APP 4. 2 How prevalent is high cholesterol? Choose an American adult at random. Define two events: A = the person has a cholesterol level of 240 milligrams per deciliter of blood (mg/dl) or above (high cholesterol) B = the person has a cholesterol level of 200 to <240 mg/dl (borderline high cholesterol) According to the American Heart Association, P(A) = 0. 16 and P(B) = 0. 29. 1. Explain why events A and B are mutually exclusive. 2. Say in plain language what the event “A or B” is. Find P(A or B). 3. Let C be the event that the person chosen has a cholesterol level below 200 mg/dl (normal cholesterol). Find P(C). Statistics and Probability with Applications, 3 rd Edition 14

LESSON APP 4. 2 How prevalent is high cholesterol? 1. Explain why events A and B are mutually exclusive. 2. Say in plain language what the event “A or B” is. Find P(A or B). 3. Let C be the event that the person chosen has a cholesterol level below 200 mg/dl (normal cholesterol). Find P(C). Statistics and Probability with Applications, 3 rd Edition 15

Basic Probability Rules Learning Targets After this lesson, you should be able to: ü Give a probability model for a chance process with equally likely outcomes and use it to find the probability of an event. ü Use the complement rule to find probabilities. ü Use the addition rule for mutually exclusive events to find probabilities. Statistics and Probability with Applications, 3 rd Edition 16