4-2 Addition Rule and Multiplication Rule 167 A tree diagram is a graph of the possible outcomes of a procedure, as in Figure 4-4. Figure 4-4 shows that if both answers are random guesses, all 10 branches are equally likely and the probability of getting the correct pair (T,c) is 1>10. For each response to the first question, there are 5 responses to the second. The total number of outcomes is 5 taken 2 times, or 10. The tree diagram in Figure 4-4 therefore provides a visual illustration for using multiplication. Summary of Addition Rule and Multiplication Rule Addition Rule for P1A or B2: The word or suggests addition, and when adding P1A2 and P1B2, we must add in such a way that every outcome is counted only once. Multiplication Rule for P1A andB2: The word and for two trials suggests multiplication, and when multiplying P1A2 and P1B2, we must be sure that the probability of event B takes into account the previous occurrence of event A. Ta Tb Tc Td Te Fa Fb Fc Fd Fe a b c d e a b c d e T F 10 5 5 2 3 FIGURE 4-4 Tree Diagram of Test Answers Statistical Literacy and Critical Thinking 1. Notation When randomly selecting a new smartphone, D denotes the event that it has a manufacturing defect. What do P1D2 and P1D2 represent? 2. Notation When randomly selecting adults, let M denote the event of randomly selecting a male and let B denote the event of randomly selecting someone with blue eyes. What does P1M B2 represent? Is P1M B2 the same as P1B M2? 3. Sample for a Poll There are 30,488,983 Californians aged 18 or older. If The Gallup organization randomly selects 1068 adults without replacement, are the selections independent or dependent? If the selections are dependent, can they be treated as being independent for the purposes of calculations? 4.Probability for Significance The drug Lipitor (atorvastatin) is used to treat high cholesterol. In a clinical trial of Lipitor, 47 of 863 treated subjects experienced headaches (based on data from Pfizer). A claim is made that fewer than 10% of subjects treated with Lipitor experience headaches. If the actual rate of headaches is 10%, there is a probability of 0.000000987 of getting a result of 47 or lower. Does 47 appear to be significantly low? Does it appear that the claim is supported by the data? Finding Complements. In Exercises 5–8, find the indicated complements. 5.Women in Movies In a recent year, speaking characters in movies were 68.2% male. What is the probability of randomly selecting a character with a speaking part and getting a female? What should be the value of that probability? 6. Smartwatch Based on a survey by Consumer Technology Association, smartwatches are used in 18% of U.S. households. Find the probability that a randomly selected U.S. household has no smartwatches. 7.Laundry Symbols Based on a New Generation of Stains survey, 13% of U.S. adults know that the care-instruction symbol ∆ on clothing means that any bleach can be used. Find the probability of randomly selecting an adult in the U.S. who does not know that. 8.Sobriety Checkpoint When the author observed a sobriety checkpoint conducted by the Dutchess County Sheriff Department, he saw that 676 drivers were screened and 6 were arrested for driving while intoxicated. Based on those results, we can estimate that P1I2 = 0.00888, where I denotes the event of screening a driver and getting someone who is intoxicated. What does P1I2 denote, and what is its value? 4-2 Basic Skills and Concepts
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