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CHAPTER 5 Probability: What Are the Chances?

CHAPTER 5 Probability: What Are the Chances?. 5.2 Probability Rules. 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.

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CHAPTER 5 Probability: What Are the Chances?

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  1. CHAPTER 5Probability: What Are the Chances? 5.2 Probability Rules

  2. 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 Sof 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. Example: Building a probability model Toss a Coin 3 Times HHH HHT HTT TTT HTH THT THH TTH Since the coin is fair, each outcome is equally likely. Each outcome has probability 1/8. Sample Space 8 Outcomes

  4. 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. 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. Basic Rules of Probability We can summarize the basic probability rules more concisely 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, • 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).

  7. 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: • is a high school graduate. • is a high school graduate and owns a home. • is a high school graduate or owns a home. 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. (a) Each student is equally likely to be chosen. 310 students are graduates. So, P(graduate) = P(A) = 310/500. (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

  8. General Addition Rule for Two Events BBBBA Event A Graduate P(A) = 310/500 Event B Homeowner P(B)= 340/500 We can’t use the addition rule for mutually exclusive events unless the events have no outcomes in common. B A Outcomes here are double-counted by P(A) + P(B) Event A and B Graduate and homeowner P(A and B) = 221/500 General Addition Rule for Two Events 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)

  9. Venn Diagrams and Probability Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed. The complement ACcontains 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. Venn Diagrams and Probability The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A andB. 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. Venn Diagrams and Probability A 89 Recall the example on high school graduates and homeowners. We can use a Venn diagram to display the information and determine probabilities. B 119 71 221 Define events A: graduate and B: homeowner.

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