So, our professor provided this solution to the problem below:

8. When a machine is improperly adjusted, it has a probability of 0.15 of producing a defective part. Each day, the machine is run until 3 defective parts are produced. a. What is the probability that the improperly adjusted machine will produce 5 parts before being stopped? Answer This is a negative binomial problem (number of good parts before a certain number of bad parts]. The easiest way to think ofthis is to consider it a binomial [2 good parts, 2 bad parts] then the last part is defective. So we simply use the binomial for the first 1: = 4- parts with p = .15 and letX = 2. Then we take this result and multiply it by p = .15 since the last part will be defective and the machine will be stopped. 50, we have P(X = 2|n = 4, p = .15) x 0.15 = 0.9880187 x 0.15 = 0.1482028