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Understanding the probability of damaged fibers in a carpet can help us determine how likely it is to find such fibers in a given area. In this scenario, we are given that each carpet fiber has a small chance of being damaged, specifically with a probability of 0.0001 per fiber. This may seem insignificant on its own, but considering a much larger area with a high number of fibers, it becomes essential to calculate the expected occurrences of damaged fibers. In the exercise, we're assessing the chance of damaged fibers across a large space, specifically 1 square yard of carpeting, which translates into 64800 fibers. Given the high number of fibers and the small chance of each being damaged, the Poisson distribution is suitable for modeling this problem. This approach allows us to approximate probabilities for events that are rare but possible in a large sample. Using this distribution, we figure out the likelihood of discovering any damaged fibers in the carpet.

When addressing the problem of damaged fibers, it's crucial to consider the assumption that each fiber's condition is independent of the others. This means that the damage occurs independently; the state of one fiber does not affect another. This assumption simplifies mathematical modeling, allowing us to use certain types of probability distributions, like the Poisson distribution used here. The independence of events is a common assumption in probability that eases calculations. It is especially practical in large-scale analyses where keeping track of every possible interaction is unwieldy. - By considering each fiber as an independent instance, we can effectively calculate the chances for the total area, since the outcome of one fiber does not interfere with another. - This makes the Poisson distribution an appropriate choice in the context, as it specifically models scenarios involving numerous independent events with very low probabilities of individual success.

Calculating probability involves using mathematical models that help estimate the likelihood of specific outcomes. In our problem, we wanted to know the probability of having one or more damaged fibers and four or more damaged fibers in a wide section of the carpet. Given the situation described, using the Poisson distribution is ideal because it helps us deal effectively with rare events across a massive number of opportunities.Let's break down the process:- First, determine the **total number of events**, which in our case is the number of fibers—64800 in a square yard.- Then, identify the **average rate** of occurrence, represented by lambda (λ), which is calculated as the total fibers multiplied by the probability per fiber (here, 64800 x 0.0001 = 6.48).- With lambda determined, further probabilities can be derived for different events using the Poisson formula: \[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \] where \( k \) represents the number of events you're interested in.To understand the probability of one or more damaged fibers, calculate the complementary probability of having none (zero damaged fibers) and subtract it from one. Similarly, to find the probability of four or more damaged fibers, you'd compute the complement of having fewer than four.