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Homework Answers 5. Independent 9. Dependent 6. Dependent 10. Independent 7. Dependent 11. Independent 8. Independent 12. Dependent

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**The Multiplication Rule**

Section 4-4

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**Independent or Dependent Events?**

Winning first prize at the national spelling bee. Your soccer team taking first place in the Conference Championship. Independent

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**Independent or Dependent Events?**

Discovering you left your car’s headlights on. Discovering that your car’s battery is dead. Dependent

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**Independent or Dependent Events?**

Randomly selecting someone who is vegan. Randomly selecting someone who has a salad for lunch. Dependent

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**How does conditional probability effect our formula for P(A and B)?**

Critical Question! How does conditional probability effect our formula for P(A and B)?

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**Applying the Multiplication Rule**

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Example Genetics Experiment Mendel’s famous hybridization experiments involved peas, like those shown in the image below. If two of the peas shown in the figure are randomly selected without replacement, find the probability that the first selection has a green pod and the second has a yellow pod. 8/14*6/13

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Example A medical researcher is evaluating pacemakers. He is going to choose two from a pool of 3 good and 2 bad pacemakers. What is the probability of choosing a good one first, followed by a bad one [without replacement]? Is this different from the probability of choosing a bad one first, followed by a good one? 3/5*2/4 = 6/20 = .300 2/5*3/4 = 6/20 = .300

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**Example Consider two randomly selected people.**

What is the probability that both people are born on the same day of the week? What is the probability that they are both born on a Monday? 7/7*1/7 1/7*1/7 = 1/49

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Whiteboards!

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Problem #1 The Wheeling Tire Company produced a batch of 5,000 tires that includes exactly 200 that are defective. If 4 tires are randomly selected for installation on a car, what is the probability that they are all good? 4800/5000*4799/4999*4798/4998*4797/4997 0.849

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**Yes, there is a very small chance that all 100 tires are good.**

Problem #2 The Wheeling Tire Company produced a batch of 5,000 tires that includes exactly 200 that are defective. If 100 tires are randomly selected for shipment to an outlet, what is the probability that they are all good? Should this outlet plan to deal with defective tires returned by consumers? (4800/5000)^100 A sample of 100 is no more than 5% of the population so we treat the events as independent. Yes, there is a very small chance that all 100 tires are good.

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Problem #3 The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a probability of working on any given morning. What is the probability that your alarm clock will not work on the morning of an important final exam? = .100 0.100

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Problem #4 The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a probability of working on any given morning. If you have two such alarms, what is the probability that they both fail on the morning of an important final? .1*.1 = .01 0.0100

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Problem #5 The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a probability of working on any given morning. With one alarm clock you have a 0.9 probability of being awakened. What is the probability of being awakened if you use two alarm clocks? 1- P(not waking up) = = .99 0.990

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Homework P #17-19, 21

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