1. The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers CHAPTER 5 Probability: What Are the Chances? 5.2 Probability Rules

2. The Practice of Statistics, 5 th Edition2 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. Descriptions of chance behavior contain two parts: The sample space S of a chance process is the set of all possible outcomes. A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome. The sample space S of a chance process is the set of all possible outcomes. A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

3. The Practice of Statistics, 5 th Edition3 Example: Building a probability model Sample Space 8 Outcomes Sample Space 8 Outcomes Since the coin is fair, each outcome is equally likely. Each outcome has probability 1/8. Since the coin is fair, each outcome is equally likely. Each outcome has probability 1/8. Toss a Coin 3 Times HHHHHTHTTTTT HTHTHT THHTTH

4. The Practice of Statistics, 5 th Edition4 Probability Models Probability models allow us to find the probability of any collection of outcomes. An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. If A is any event, we write its probability as P(A). In the penny tossing example, suppose we define event A as “2 heads.” There are 3 outcomes that result in 2 heads. Since each outcome has probability 1/8, P(A) = 3/8. Suppose event B is defined as “not 2 heads.” What is P(B)? P(B) = 1 – 3/8 = 5/8

5. The Practice of Statistics, 5 th Edition5 Basic Rules of Probability The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities whose sum is exactly 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula The probability that an event does not occur is 1 minus the probability that the event does occur. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. 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.

6. The Practice of Statistics, 5 th Edition6 Basic Rules of Probability We can summarize the basic probability rules more concisely in symbolic form. 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, 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). Basic Probability Rules

7. The Practice of Statistics, 5 th Edition7 Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that makes probability calculations easier. What is the relationship between educational achievement and home ownership? Find the probability that an adult: (a)is a high school graduate. (b)is a high school graduate and owns a home. (c)is a high school graduate or owns a home. (a) Each student is equally likely to be chosen. 310 students are graduates. So, P(graduate) = P(A) = 310/500. Define events A: is a graduate and B: owns a home. (b) We want to find P(graduate and home owner), that is, P(A and B). Look at the intersection of the “Homeowner” row and “graduate” column. There are 221 graduates who are homeowners. So, P(A and B) = 221/500. (c) We want to find P(graduate or homeowner), that is, P(A or B). There are 310 graduates and 340 homeowners. However, 221 graduates own homes – don’t count them twice! P(A or B) = (221 + 89 + 119)/500. So, P(A or B) = 429/178 HS graduate Not HS graduate Total Homeowner221119340 Not homeowner 8971160 Total310190500

8. The Practice of Statistics, 5 th Edition8 General Addition Rule for Two Events We can’t use the addition rule for mutually exclusive events unless the events have no outcomes in common. If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) – P(A and B) General Addition Rule for Two Events Event A and B Graduate and homeowner P(A and B) = 221/500 BBBBA Event A Graduate P(A) = 310/500 Outcomes here are double-counted by P(A) + P(B) A B Event B Homeowner P(B)= 340/500

9. The Practice of Statistics, 5 th Edition9 Venn Diagrams and Probability Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed. The complement A C contains exactly the outcomes that are not in A. The events A and B are mutually exclusive (disjoint) because they do not overlap. That is, they have no outcomes in common.

10. The Practice of Statistics, 5 th Edition10 Venn Diagrams and Probability 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. Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.

11. The Practice of Statistics, 5 th Edition11 Venn Diagrams and Probability Recall the example on high school graduates and homeowners. We can use a Venn diagram to display the information and determine probabilities. Define events A: graduate and B: homeowner. Region in Venn diagramIn wordsIn SymbolsCount In the intersection of two circles Graduate and homeowner A∩B221 Inside Circle A, outside Circle B Graduate but not a homeowner A∩B c 89 Inside circle B, outside circle A Homeowner but not a graduate A c ∩B119 Outside of both circlesNot a graduate or homeowner A c ∩B c 71 B 119 A 89 71 221 HS graduate Not HS graduate Total Homeowner221119340 Not homeowner 8971160 Total310190500