In bottle production, bubbles that appear in the glass are considered defects. Any bottle that has more than two bubbles is classified as “nonconforming” and is sent to recycling. Suppose that a particular production line produces bottles with bubbles at a rate of 1.1 bubbles per bottle. Bubbles occur independently of one another.
a. What is the probability that a randomly chosen bottle is nonconforming?
b. Bottles are packed in cases of 12. An inspector chooses one bottle from each case and examines it for defects. If it is nonconforming, she inspects the entire case, replacing nonconforming bottles with good ones. This process is called rectification. If the chosen bottle conforms (has two or fewer bubbles), then she passes the case. In total, 20 cases are produced. What is the probability that at least 18 of them pass?
c. What is the expected number of nonconforming bottles in the20 cases after they have been inspected and rectified using the scheme described in part b?