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60% of homeowners in Northern Virginia have a weekly lawn service. Those homes with a lawn service, 55% have a sprinkler system. For those without a lawn service, 30% have a sprinkler system. (Note: you do NOT need to simplify these answers. However, you MUST express each in terms of the events/probability notation – e.g., P(A|B) =...) a. Draw a tree diagram to model the problem b. What is the probability that a randomly selected home has a sprinkler system? c. What is the probability that a randomly selected home with a sprinkler system has a lawn service?

60% of homeowners in Northern Virginia have a weekly lawn service. Those homes with a lawn service, 55% have a sprinkler system. For those without a lawn service, 30% have a sprinkler system. (Note: you do NOT need to simplify these answers. However, you MUST express each in terms of the events/probability notation – e.g., P(A|B) =...) a. Draw a tree diagram to model the problem b. What is the probability that a randomly selected home has a sprinkler system? c. What is the probability that a randomly selected home with a sprinkler system has a lawn service?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Problem 4: Probability in Northern Virginia Lawn Services #### Scenario: - 60% of homeowners in Northern Virginia have a weekly lawn service. - Of those with a lawn service, 55% have a sprinkler system. - Of those without a lawn service, 30% have a sprinkler system. **Note:** Express each answer in terms of the probability notation (e.g., P(A|B)). #### Tasks: a. **Draw a Tree Diagram** Model the problem using a tree diagram to represent the sequential probabilities of having a lawn service and sprinkler system. b. **Probability of Sprinkler System** - **Task:** Calculate the probability that a randomly selected home has a sprinkler system. - **Notation:** Use the law of total probability to find P(Sprinkler). c. **Probability of Lawn Service Given Sprinkler System** - **Task:** Determine the probability that a randomly selected home with a sprinkler system also has a lawn service. - **Notation:** Calculate P(Lawn Service | Sprinkler). d. **Probability of Sprinkler System Given Lawn Service** - **Task:** Find the probability that a randomly selected home with a lawn service has a sprinkler system. - **Notation:** Calculate P(Sprinkler | Lawn Service). e. **Probability of Having a Lawn Service** - **Task:** Determine the probability that a randomly selected home has a lawn service. - **Notation:** P(Lawn Service). f. **Probability of Sprinkler System Without Lawn Service** - **Task:** Calculate the probability that a randomly selected home that does not have a lawn service has a sprinkler system. - **Notation:** P(Sprinkler | No Lawn Service).
Transcribed Image Text:### Problem 4: Probability in Northern Virginia Lawn Services #### Scenario: - 60% of homeowners in Northern Virginia have a weekly lawn service. - Of those with a lawn service, 55% have a sprinkler system. - Of those without a lawn service, 30% have a sprinkler system. **Note:** Express each answer in terms of the probability notation (e.g., P(A|B)). #### Tasks: a. **Draw a Tree Diagram** Model the problem using a tree diagram to represent the sequential probabilities of having a lawn service and sprinkler system. b. **Probability of Sprinkler System** - **Task:** Calculate the probability that a randomly selected home has a sprinkler system. - **Notation:** Use the law of total probability to find P(Sprinkler). c. **Probability of Lawn Service Given Sprinkler System** - **Task:** Determine the probability that a randomly selected home with a sprinkler system also has a lawn service. - **Notation:** Calculate P(Lawn Service | Sprinkler). d. **Probability of Sprinkler System Given Lawn Service** - **Task:** Find the probability that a randomly selected home with a lawn service has a sprinkler system. - **Notation:** Calculate P(Sprinkler | Lawn Service). e. **Probability of Having a Lawn Service** - **Task:** Determine the probability that a randomly selected home has a lawn service. - **Notation:** P(Lawn Service). f. **Probability of Sprinkler System Without Lawn Service** - **Task:** Calculate the probability that a randomly selected home that does not have a lawn service has a sprinkler system. - **Notation:** P(Sprinkler | No Lawn Service).
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