91Ó°ÊÓ

Understanding statistical outcomes is essential for analyzing probability events. When we refer to statistical outcomes, we're talking about all the possible results that can occur from a probabilistic event. In the context of our exercise, the statistical outcomes are the distinct pairs of computer boards that can be selected for inspection from a lot of five boards.

Each pair is a potential outcome that needs to be considered when calculating probabilities. In our example, there are 10 different possible statistical outcomes: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), and (4,5). To ensure we accurately understand the problem, it's vital to list all these outcomes clearly as it forms the basis for further calculations involving probability.

The probability of picking a defective board is a practical problem that can occur in manufacturing processes. It involves analysis of random events to determine the likelihood of selecting a defective item from a batch. In our exercise, two boards out of five are defective, and we want to know the probability of picking none, one, or both of these defective boards when two are randomly selected for inspection.

To improve the student's comprehension of the concept, it's crucial to walk through the probability calculations for each scenario. For instance, we found the probability of selecting no defective boards (\(x = 0\)) to be \frac{3}{10}, since there are 3 outcomes without defective boards. Similarly, the calculation for selecting exactly one defective board (\(x = 1\)) yields a probability of \frac{6}{10}, and selecting both defective boards (\(x = 2\)) has a probability of \frac{1}{10}. This step-by-step approach aids in reinforcing the student's understanding of probabilities in discrete events.

Combinatorics is the branch of mathematics dealing with the study of counting, combination, and permutation of sets. It's a powerful tool for calculating the number of possible outcomes in problems involving probability. We applied combinatorics in our exercise to determine the 10 possible outcomes for choosing pairs of boards.

In more complex problems, combinatorial formulas such as the factorial, permutations, and combinations come into play. These formulas are essential for calculating the number of ways that events can occur, which in turn helps us to compute probabilities. For example, in our exercise, we intuitively listed all pairs, but in larger sets, combinatorial formulas would be necessary to avoid errors and to efficiently calculate the total number of possible outcomes without listing them all.