Question 2 (20 points) ina In a shipment of 20 engines, history shows that the probability of any one engine proving unsatisfactory is 0.1 a. The probability that all 20 engines are satisfactory: (5 points) #INSERT YOUR ANSWER HERE R b. Use the Binomial approximation to calculate the probability that more than 10 engines are defective? (5 points) #INSERT YOUR ANSWER HERE c. Use the binomial approximation to calculate the probability that at most three engines are defective? (5 points) #INSERT YOUR ANSWER HERE d. What is the probability that the second engine is defective given the first engine is not defective? From the result, draw the conclusion if the first and second engines are dependent or independent. (5 points) #INSERT YOUR ANSWER HERE Question 3 (25 points) On average, John goes to his parents' place twice a week for visits. a. Find the probabilities that John goes to his parents' place 0 to 5 times in a week? (5 points) #INSERT YOUR ANSWER HERE b. Find the probability that John goes to his parents' place 3 times or less in a week? (5 points) #INSERT YOUR ANSWER HERE c. Compare the similarity between Binomial and Poisson distribution. (15 points @ 5 point each) 1. Create 100,000 samples for a Binomial random variable using parameters described in Question 2 2. Create 100,000 samples for a Poisson random variable using parameters described in Question 3 3. then illustrate on how well the Poisson probability distribution approximates the Binomial probability distribution. (BONUS) HINT: use multhist() from the 'plotrix' package #INSERT YOUR ANSWER HERE