Final answer:
The exact probability that an item is defective cannot be calculated without specific data on the number of defects and items inspected. Probability can be calculated using formulas if the defect rate is known, potentially with the aid of calculators and considering distribution rules like the 68-95-99.7 empirical rule.
Step-by-step explanation:
To calculate the probability that an item is defective, given complete inspection, we require specific information about the number of defects found during the inspection. However, since the question does not provide sufficient data, it's not possible to calculate an exact probability without additional context. Hypothetically, if we knew the total number of items and identified defective items, we could use the formula for probability: Probability of a defective item = (number of defective items) / (total number of items).
In the context of the provided examples from Try It Σ and Example 4.19, if the probability of an item (like a steel rod or a DVD player) being defective is known, we can calculate more specific probabilities. For instance, the probability that the first defect occurs on the ninth steel rod can be calculated using the geometric distribution formula P(x = 9) = (1-p)8 × p, where p = 0.01 in this case. The probability that, in a sample of 12 DVD players chosen from a shipment with a known number of defects, at most two are defective can be computed using the hypergeometric distribution or binomial approximation depending on the scenario. These calculations can often be aided by the use of a TI-83/84 calculator.
When taking random samples from a production line or a shipment with a particular defect rate, we can also apply statistical rules such as the 68-95-99.7 empirical rule if the distribution of defects follows a normal distribution. This rule gives us a way to predict the spread of defects around the mean in a population.