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Quality control is a crucial process used by manufacturers to ensure the products meet certain standards or specifications. This often involves the inspection and testing of materials, components, and products to identify defects.
For example, in the context of a computer manufacturer, a lot of computer boards might be inspected to detect any defective pieces. By checking a subset of the total boards—through random sampling or specified criteria—manufacturers can infer the overall quality of the whole batch.
In our given problem, the manufacturer does quality control by inspecting `two` boards out of a lot of `five`. Identifying defects early can save costs and maintain product standards, which is vital to customer satisfaction and brand reputation.
This method helps determine whether further action is needed: whether all boards should be rejected or passed on for further processing. Having a robust quality control process ensures that only defect-free products make it to the customer, safeguarding the brand's reputation.

The concept of combinations is essential when it comes to selecting items from a group where the order does not matter. Combinations can be calculated using the formula \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to select.
In the problem scenario, we are asked to choose 2 boards out of 5 for quality inspection. Using the combinations formula, \( \binom{5}{2} \), we find there are 10 possible ways to pick any two boards from the lot of five. This calculation is done as follows:

• \( n = 5 \), \( k = 2 \)
• \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \)

Each combination represents one unique selection outcome, and understanding how to calculate these outcomes is vital in determining probabilities in quality control scenarios like this one.

A random variable is a numerical outcome of a random process. It assigns a number to each possible outcome in a sample space. In our exercise, the random variable \( X \) represents the number of defective boards found in a sample of two boards.
Random variables can be discrete or continuous. Here, \( X \) is a discrete random variable because it can take on a countable number of values (0, 1, or 2, in this case).
Given that boards 1 and 2 are the defective ones, each pair chosen from our list could result in one of the following for \( X \):

• \( X = 0 \): No defective boards (e.g., pairs like (3,4))
• \( X = 1 \): One defective board (e.g., pairs like (1,3))
• \( X = 2 \): Both boards are defective (e.g., pair (1,2))

Calculating these probabilities gives insight into the likelihood of each defect scenario occurring, which informs the decision-making process related to quality control.

The cumulative distribution function (CDF) of a random variable \( X \) provides the probability that \( X \) will take a value less than or equal to \( x \). It's a way of summarizing the probability distribution of a random variable.
For our exercise, the CDF \( F(x) \) of \( X \) is determined by calculating cumulative probabilities:

• \( F(0) = P(X \leq 0) = P(X = 0) = \frac{3}{10} \)
• \( F(1) = P(X \leq 1) = P(X = 0) + P(X = 1) = \frac{3}{10} + \frac{6}{10} = \frac{9}{10} \)
• \( F(2) = P(X \leq 2) = 1 \), as \( X \) can take on values up to 2

The CDF gives us a complete picture of the probability of observing different numbers of defects, helping manufacturers gauge the overall quality from the inspection and decide on further actions.