Part 1: Discrete Probability Distributions 1. You work with the office of city planning which is currently evaluating a new contract for a major highway. The chosen contractor claims that the roads they build usually have a single (1) defect per 50 miles of road, assume this follows a Poisson distribution. The portion of highway that your city is building is 30 miles long. a. What is the probability that there are no (0) defects in the completed highway? b. Within the contract there is a clause which states that if 5 or more defects are found in the completed project then the contractor must reimburse a third of the final payment. What is the probability that this happens? 2. Loot boxes are a videogame consumable that a player can purchase with real money and redeem for a set of randomized virtual items. These have been highly criticized in recent years due to their gambling nature along with the unknown workings behind the randomization of the items. You are an avid player of Roverwatch, a team-based shooter game where all characters are dogs. In this game you can open loot boxes in exchange for random character outfits that range in level according to their complexity, legendary being the best. The game’s developer states that there is an 8% chance (0.08 probability) of getting a legendary item in any loot box however you are unsure about this. You open 200 loot boxes and don’t get a single legendary item until your 26th loot box. Additionally, only 7 of the 200 loot boxes contained a legendary item. a. What is the probability of both these events (first legendary being in the 26th loot box, only 7 of the 200 boxes containing a legendary)? b. Do you think the game’s developer is being honest in stating the probability of a legendary item? Why? Hint: Consider the z-scores or relative location from numerical summary stats. Econ 3400 HW3 Spring 2020 2 Part 2: Continuous Probability Distributions 1. Draw clear graphs of the following, each letter below should be a single separate graph with proper labels where required (everything from each prompt should be clearly visible in each graph): a. Two normal distributions, both with mean 10, one with standard deviation k and the other with standard deviation 3k (both distributions should appear on the same graph). b. Three normal distributions, all with standard deviation ??, one with mean 0, one with mean 10, one with mean 30 (all three distributions should appear on the same graph). c. A standard normal distribution where the area that represents the probability of our variable being between 1 and 2 (??(1 = ?? = 2)) is shaded. d. A standard normal distribution where the area that represents the probability of our variable being greater than 3 (??(?? = 3)) is shaded. 2. COTA states that on average a bus comes to your stop every 15 minutes. You have just arrived at the bus stop and realize that you left your umbrella in your apartment, it will take you 10 minutes to go back for it and you decide to do this since it is supposed to snow today. Assume bus arrivals are exponentially distributed. a. What is the probability that a bus arrives while you are going back for your umbrella? After coming back to the bus stop with your umbrella, you realize that you’re late to Econ 3400 and don’t want to miss a second of it. Aware of how long it takes you to walk from the bus stop where you get off to class you know that if the bus takes more than 20 minutes to get here you will be late. b. What is the probability that you are late to Econ 3400? c. What does the memoryless property imply in this setting of waiting for a bus?
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