Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities.
A glass manufacturer finds that 1 in every 1000 glass items produced is warped. Find the probability that (a) the first warped glass item is the 12 th item produced, (b) the first warped item is the first, second, or third item produced, and (c) none of the first 10 glass items produced are defective.
(a) $P$ (the first warped glass item is the 12 th item produced) = (Round to three decimal places as needed.)

Step 1 ：The problem is asking for the probability of a certain event happening at a specific trial in a sequence of independent trials. This is a classic case of a geometric distribution problem. The geometric distribution is used to model the number of trials it takes for an event to occur for the first time. In this case, the event is the production of a warped glass item, and the trials are the production of each glass item.

Step 2 ：The probability of producing a warped glass item is 1 in 1000, or 0.001.

Step 3 ：The probability that the first warped glass item is the 12th item produced is given by the formula for the geometric distribution: \(P(X = k) = (1 - p)^{(k - 1)} * p\) where p is the probability of success on each trial, k is the number of trials, and X is the random variable representing the number of trials until the first success. In this case, p = 0.001 and k = 12.

Step 4 ：Substituting the given values into the formula, we get: \(P(X = 12) = (1 - 0.001)^{(12 - 1)} * 0.001\)

Step 5 ：Solving the above expression, we get the probability as approximately 0.001, or 0.1%. This seems like a very low probability, which makes sense given that only 1 in every 1000 glass items is warped.

Step 6 ：Final Answer: The probability that the first warped glass item is the 12th item produced is approximately \(\boxed{0.001}\).