1) Discrete Probability Distributions

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 the 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. Over the 200 loot boxes that you opened, how many would you have expected to contain a legendary item? (I.e. Expected value)

b. What is the probability that only 7 of the 200 boxes contain a legendary?

c. What is the probability that the first legendary comes in the 26th loot box?