MATH 314 Homework 05

Due 2026-10-01 by 11:59pm

1. A software development team is working on improving the response time of an algorithm. Currently, when given random inputs, the algorithm successfully executes within the desired response time in 92% of cases. Suppose the team runs the algorithm with 100 random inputs as a testing phase.

   a. What is the probability that the algorithm will execute within the desired response time for exactly 95 out of the 100 inputs?

   b. What is the probability that the algorithm will execute within the desired response time for 90 or fewer of the 100 inputs?

   c. What is the probability that between 80 and 95, inclusive of both 80 and 95, of the 100 inputs execute within the desired response time?

   d. On average, in how many cases do you expect the algorithm to successfully execute within the desired response time?

2. In certain remote habitats, rare species can be difficult to observe. Ecologists often rely on random sampling to estimate population densities and study their behavior. Assume, on average, 4.4 sightings of a rare bird species are recorded every week in the wild.

   a. What is the probability that in a given week, there will be exactly 3 sightings of this bird species?

   b. What is the probability that in a given week, there will be 5 or more sightings?

   c. After a reforestation initiative, the number of sightings increases to an average of 7.2 per week. If an ecologist spends a week observing, what is the probability that they will record at least one sighting of this bird species?