MATH 350 Homework 06

Throughout, please use notation like, but not necessarily equivalent to, $X \sim D(...)$ where $D$ is some distribution, and $\mathbb{P}[X \le x]$, as appropriate to each questions subpart. Also, please provide answers in mathematical symbols.

1. The airspace around a major airport is a hive of activity. Air traffic controllers monitor the arrival and departure of aircraft, ensuring safe and efficient operations. Suppose that the air traffic control at a particular airport has noticed that the number of aircraft approaching its airspace for landing follows a Poisson distribution. On average, 15 aircraft approach the airport's airspace for landing every hour.
   1. What is the probability that exactly 10 aircraft approach the airport in a given hour?
   2. What is the probability that more than 20 aircraft approach the airport in a given hour?
   3. During peak hours, the average number of approaching aircraft increases to 25 per hour. What is the probability that between 20 and 30 aircraft (inclusive) approach the airport during a peak hour?
2. A large agricultural field is being monitored for pest infestation. Over the course of a growing season, it's observed that the number of pest-infested plants in randomly selected square-meter plots follows a Poisson distribution. On average, 5 plants are found to be pest-infested in each square-meter plot.
   1. What is the probability that a randomly selected square-meter plot has exactly 3 pest-infested plants?
   2. What is the probability that a randomly selected square-meter plot has no pest-infested plants at all?
3. In microbiological studies, Petri dishes are often inoculated with a diluted sample to count the number of bacteria present in a specimen. After allowing for growth, researchers count the number of bacterial colonies that form on the dish. The number of colonies that appear on a Petri dish is assumed to follow a Poisson distribution. Suppose that, on average, 12 bacterial colonies are observed per Petri dish when using a specific dilution method.
   1. What is the probability that a randomly selected Petri dish will show exactly 10 bacterial colonies?
   2. What is the probability that a randomly selected Petri dish will show 15 or more bacterial colonies?
   3. Given that a certain antibiotic was added to another set of Petri dishes, the average number of bacterial colonies reduced to 6 per dish. What is the probability that a dish from this set will have 3 or fewer bacterial colonies?