MATH 350 Homework 05

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

  1. A mechanical engineer is testing the reliability of a new type of bolt used in aerospace applications. In a sample test of 50 bolts, each bolt has a 98% chance of withstanding a force of 2000N without breaking.
    1. What is the probability that exactly 49 bolts withstand the force?
    2. What is the probability that at least 48 bolts withstand the force?
    3. What is the probability that no more than 10 bolts withstand the force?
    4. If the company decides to use these bolts in a batch of 100 in a real-world application, what is the expected number of bolts that will withstand the force, E[X]\mathbb{E}[X]? What is the variance, V[X]\mathbb{V}[X]?
  2. 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.
    1. What is the probability that the algorithm will execute within the desired response time for exactly 95 out of the 100 inputs?
    2. What is the probability that the algorithm will execute within the desired response time for 90 or fewer of the 100 inputs?
    3. What is the expected number of successful executions, E[X]\mathbb{E}[X]? What is the variance, V[X]\mathbb{V}[X]?
  3. Suppose an email message fails to send for 1 out of every 10000 emails you send.
    1. What's the probability that 1 out of the next 500 emails of yours fails to send?
    2. What's the probability that at most 1 out of the next 500 emails of yours fails to send?
    3. What's the probability that at least 2 out of the next 500 emails of yours fail to send?
    4. What is the expected number of emails that fail to send, E[X]\mathbb{E}[X]? What is the variance, V[X]\mathbb{V}[X]?