Source: An Introduction to Statistical Methods and Data Analysis (R Lyman Ott, Michael Longnecker)
Here's an example from the text which introduces a binomial experiment and asks us to determine whether or not the experiment satisfies all the requirements of a binomial test.
A large power utility company uses gas turbines to generate electricity. The engineers employed at the company monitor the reliability of each turbine—that is, the probability that the turbine will perform properly under standard operating conditions over a specified period of time. The engineers wanted to estimate the probability a turbine will operate successfully for 30 days after being put into service. The engineers randomly selected 75 of the 100 turbines currently in use and examined the maintenance records. They recorded the number of turbines that did not need repairs during the 30-day time period. Is this a binomial experiment?
Now, the author determines that this case does not satisfy all requirements of a binomial experiment. Specifically, that the probability of success does not remain constant from trial to trial. I do not understand his conclusion about that, however.
- Is the probability of success the same from trial to trial? No. If we let success denote a turbine “did not need repairs,” then the probability of success can change considerably from trial to trial. For example, suppose that 15 of the 100 turbines needed repairs during the 30-day inspection period. Then p, the probability of success for the first turbine examined, would be 85/100 .85. If the first trial is a failure (turbine needed repairs), the probability that the second turbine examined did not need repairs is 85/99 .859. Suppose that after 60 turbines have been examined, 50 did not need repairs and 10 needed repairs. The probability of success of the next (61st) turbine would be 35/40 .875.
Specifically, why is the author using a posteriori knowledge about the overall probabilities (15/100) to determine that the probability success and failure of the turbines is not constant during the experiment? It seems to me that if he has no reason to assume that any turbine will be in need of repair, then the probabilities of success should be the same for each.