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

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Chapter 5: Probability: What Are the Chances? Section 5.2 Probability Rules Pierced Pierced Ears + Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules Two-Way Tables and Probability + Probability Rules When finding probabilities involving two events, a two-way table can display the sample space in a way that makes probability calculations easier. Consider the example on page 303. Suppose we choose a student at random. Find the probability that the student (a) has pierced ears. (b) is a male with pierced ears. (c) is a male or has pierced ears. Define events A: is male and B: has pierced ears. (a) Each student is equally likely to be chosen. 103 students have pierced ears. So, P(pierced ears) = P(B) = 103/178. (b) We want to find P(male and pierced ears), that is, P(A and B). Look at the intersection of the “Male” row and “Yes” column. There are 19 males with pierced ears. So, P(A and B) = 19/178. (c) We want to find P(male or pierced ears), that is, P(A or B). There are 90 males in the class and 103 individuals with pierced ears. However, 19 males have pierced ears – don’t count them twice! P(A or B) = (19 + 71 + 84)/178. So, P(A or B) = 174/178 Two-Way Tables and Probability + Probability Rules Note, the previous example illustrates the fact that we can’t use the addition rule for mutually exclusive events unless the events have no outcomes in common. The Venn diagram below illustrates why. General Addition Rule for Two Events + Probability Rules 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) or P(A U B) = P(A) + P(B) – P(A ∩ B) + Checkpoint A standard deck of playing cards (with joker removed) consists of 52 cards in four suits – clubs, diamonds, hearts and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jack, queen, and king are referred to as “face cards.” Imagine that we shuffle the deck thoroughly and deal one card. Let’s define events A: getting a face card and B: getting a heart. 1) Make a two-way table that displays the sample space. Face Card Non-face card Heart 3 10 Non-heart 9 30 + Checkpoint 2) Find P(A and B) 3 P(face card and heart) 52 3) Explain why P(A or B) ≠ P(A) + P(B). Then use the general addition rule to find P(A or B). Since A and B are not disjoint, this is not equal to the sum of the probability of A and the probability of B. P(A or B) = P(A) + P(B) – P(A and B) = 0.423 Venn Diagrams and Probability + Probability Rules Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and notation have been developed. The complement AC 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. Venn Diagrams and Probability + Probability Rules 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. Venn Diagrams and Probability + Probability Rules Recall the example on gender and pierced ears. We can use a Venn diagram to display the information and determine probabilities. Define events A: is male and B: has pierced ears. + Example – Venn Diagrams, Two- Way Tables and Probability According to the National Center for Health Statistics, in December 2008, 78% of U.S. households had a traditional landline telephone, 80% of households had cell phones, and 60% had both. Suppose we randomly selected a household in December 2008. 1) Make a two-way table that displays the sample space of this chance process. Cell Phone No Cell Phone Total Landline 0.60 0.18 0.78 No Landline 0.20 0.02 0.22 Total 0.80 0.20 1.00 + 2) Construct a Venn Diagram to represent the outcomes of this chance process. + 3) Find the probability that the household has at least one of the two types of phones. To find the probability that the household has at least one of the two types of phones, we need to find the probability that the household has a landline, a cell phone, or both. P(A U B) = P(A) + P(B) – P(A ∩ B) = 0.78 + 0.80 – 0.60 = 0.98 There is a 98% chance that the household has at least one of the two types of phones. 4) Find the probability that the household has a cell phone only. P(cell phone only) P(Ac B) 0.20 + Calculators & Probability Many probability problems involve simple computations that you can do on your calculator. It may be tempting to just write down your final answer without show the supporting work. Don’t do it!!! A “naked answer,” even if it’s correct, will usually earn you no credit on a free-response question. + Section 5.2 Probability Rules Summary In this section, we learned that… A probability model describes chance behavior by listing the possible outcomes in the sample space S and giving the probability that each outcome occurs. An event is a subset of the possible outcomes in a chance process. For any event A, 0 ≤ P(A) ≤ 1 P(S) = 1, where S = the sample space number of outcomes corresponding to event A S P(A) If all outcomes in are equally likely, total number of outcomes in sample space P(AC) = 1 – P(A), where AC is the complement of event A; that is, the event that A does not happen. + Section 5.2 Probability Rules Summary In this section, we learned that… Events A and B are mutually exclusive (disjoint) if they have no outcomes in common. If A and B are disjoint, P(A or B) = P(A) + P(B). A two-way table or a Venn diagram can be used to display the sample space for a chance process. The intersection (A ∩ B) of events A and B consists of outcomes in both A and B. The union (A ∪ B) of events A and B consists of all outcomes in event A, event B, or both. The general addition rule can be used to find P(A or B): P(A or B) = P(A) + P(B) – P(A and B).
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