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MAT 155­DY1 and DY2

Section 4.4 Multiplication Rule ­ Basics

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Section 4­4 Multiplication Rule ­ Basics173/5. Classify the two events as independent or dependent:(a) Randomly selecting a quarter made before 2001. Randomly selecting a second quarter made before 2001.(b) Randomly selecting a TV viewer who is watching The Barry Manilow Biography. Randomly selecting a second TV viewer who is watching The Barry Manilow Biography.(c) Wearing plaid shorts with black sox and sandals. Asking someone on a date and getting a positive response.

Classifying events as independent or dependent may be debatable. However, as a class we classified (a) and (b) as Independent and (c) as Dependent.

We assumed that there were 100 quarter total with 42 of them made before 2001 and 58 after 2001. Thus, P(B) = 42/100 = 0.42.This led to the discussion that if we use replacement the events would be independent. If we did not use replacement, the events would be dependent. So problems like these may be subject to interpretation.

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173/6.Classify the two events as independent or dependent: (See media Events & Independence.)(a) Find that your calculator is working. Finding that your cell phone is working.(b) Finding that your kitchen toaster is not working. Finding that your refrigerator is not working.(c) Drinking or using drugs until your driving ability is impaired. Being involved in a car crash.

In (b), the toaster and refrigerator are independent of each other. However, the fact that they are not working means that the “not working” aspect is probably dependent, especially if they are both on the same electrical circuit.We chose to think of them on the same circuit, although a properly wired house would have them on different circuits.

Section 4­4 Multiplication Rule ­ Basics

173/6. Classify the two events as independent or dependent: (See media Events & Independence in CourseCompass.)

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173/8. Computer password of two characters. Random letter for first and a digit for the second. What is probability that password is "K9"? Would this password be effective against hacker?

There are 26 letters and 10 digits. Thus, P(K) = 1/26 and P(9) = 1/10.Choosing the letter K and then the digit 9 are independent because the choice of K does not affect the probability of choosing the 9. For two independent events, the probability of both equals the product of the two probabilities.

Section 4­4 Multiplication Rule ­ Basics

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173/9. Among 123 hunters injured when mistaken for game, 6 were wearing orange. If a follow­up study begins with the random selection of hunters from this sample of 123, find the probability that the first two selected hunters were both wearing orange.(a) Assume that the first hunter is replaced before the next one is selected.(b) Assume that the first hunter is not replaced before the second one is selected.(c) Given a choice between selecting with replacement and selecting without replacement, which choice makes more sense in this situation? Why?

Section 4­4 Multiplication Rule – Basics (See Applying the Multiplication Rule.)

We need to note with or without replacement before calculating the probability.• P(A and B) = 0.002379536 may be rounded to 0.002.(b) P(A and B) = 0.0019992003, rounded would be 0.002.NOTE: This shows that with a large number of possible outcomes dependent events may be treated as independent with little loss in accuracy.

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174/10. In the 108th Congress, the Senate consists of 51 Republicans, 48 Democrats, and 1 Independent. If a lobbyist for the tobacco industry randomly selects three different Senators, what is the probability that they are all Republicans? Would a lobbyist be likely to use random selection in this situation?

P(all three Republican) = (51/100)(50/99)(49/98) = 0.1287878 … = 0.129 rounded to 3 places.No, the Senators would probably be carefully selected to find those supportive of the tobacco industry.

Section 4­4 Multiplication Rule ­ Basics

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174/12. Use a 95% confidence level. If six different organizations conduct independent polls, what is the probability that all six of them are accurate within the claimed margins of error? Does the result suggest that with a confidence level of 95%, we can expect that almost all polls will be within the claimed margin of error?

Notice two key words in this problem: different and independent. Choosing different events usually implies dependent; however, this problem states that they are independent.

Section 4­4 Multiplication Rule ­ Basics

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175/17. If two different pedestrian deaths are randomly selected, find the probability that they both involved intoxicated drivers.

Choosing two different pedestrian deaths makes this dependent events. P(ID and ID) = 0.0195060047 = 0.020 to 3 places.

Section 4­4 Multiplication Rule ­ Basics

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175/18. If two different pedestrian deaths are randomly selected, find the probability that they both involved intoxicated pedestrians.

Choosing two different pedestrian deaths makes this dependent events. P(IP and IP) = 0.1086418225 = 0.109 to 3 places.

Section 4­4 Multiplication Rule ­ Basics

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175/19. Pedestrian Deaths(a) If one of the pedestrian deaths is randomly selected, what is the probability that it involves a case in which neither the pedestrian nor the driver was intoxicated?(b) If two different pedestrian deaths are randomly selected, what is the probability that in both cases, neither the pedestrian nor the driver was intoxicated?

Part (a) is dependent because without replacement; (c) is independent because of replacement.Answers to 3 places: (a) 0.590 (b) 0.348 (c) 0.348NOTE: For large samples, dependent events (b) may be treated as independent events (c).

Section 4­4 Multiplication Rule ­ Basics

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175/19. Pedestrian Deaths(c) If two pedestrian deaths are randomly selected with replacement, what is the probability that in both cases, neither the pedestrian nor the driver was intoxicated? (d) Compare the results from parts (b) and (c).

Part (a) is dependent because without replacement; (c) is independent because of replacement.Answers to 3 places: (a) 0.590 (b) 0.348 (c) 0.348NOTE: For large samples, dependent events (b) may be treated as independent events (c).

Section 4­4 Multiplication Rule ­ Basics

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175/20. Pedestrian Deaths(a) If one of the pedestrian deaths is randomly selected, what is the probability that it involves an intoxicated pedestrian and an intoxicated driver?(b) If two different pedestrian deaths are randomly selected, what is the probability that in both cases, both the pedestrian and the driver were intoxicated?

Part (a) is dependent because without replacement; (c) is independent because of replacement.Answers to 3 places: (a) 0.060 (b) 0.004 (c) 0.004NOTE: For large samples, dependent events (b) may be treated as independent events (c).

Section 4­4 Multiplication Rule ­ Basics

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175/20. Pedestrian Deaths(c) If two pedestrian deaths are randomly selected with replacement, what is the probability that in both cases, both the pedestrian and the driver were intoxicated? (d) Compare the results from parts (b) and (c).

Part (a) is dependent because without replacement; (c) is independent because of replacement.Answers to 3 places: (a) 0.060 (b) 0.004 (c) 0.004NOTE: For large samples, dependent events (b) may be treated as independent events (c).

Section 4­4 Multiplication Rule ­ Basics