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The Poisson distribution is a vital tool in probability, especially when dealing with events that happen independently. It's often used to model the number of times an event occurs within a specific interval. For example, in this carpet fiber problem, we're examining how many fibers get damaged. When the probability of an individual event happening is low (like damaging a fiber) and you have a large number of trials (the total fibers), the Poisson distribution becomes handy. To use the Poisson distribution, you need a parameter \( \lambda \), which represents the average number of events (or damages, in this context) in the interval. In this problem, \( \lambda = 6.48 \), revealing damage occurs, on average, about 6.48 times in a square yard.

The expected value (`EX`) is simply the mean or average number of times an event should occur based on probabilities. For scenarios involving large sample sizes and small probabilities of success, like the damaged fibers, calculating the expected value can guide us on what to reasonably anticipate.In our case, the expected value of damaged fibers is found by multiplying the total fibers present by the probability of any single fiber being damaged. Thus, we have \[64800 \times 0.0001 = 6.48\]This tells us that, on average, we should expect about 6.48 damaged fibers in one square yard.

In probability, approximating allows us to make sense of complex situations. Using the Poisson distribution to approximate probabilities is common when dealing with many independent trials and low success probabilities. Steps involved in approximating:- Convert units to calculate total items or trials
- Compute the expected number of events
- Model with the Poisson distribution using \( \lambda \) as the expected number of eventsIn our exercise, the probabilities of specific scenarios (like 1 damaged fiber, or 4 or more) are approximated to make calculations feasible without direct enumeration of every outcome. This method simplifies the process significantly.

Independent events in probability mean that the occurrence of one event does not affect the occurrence of another. In the world of probability, the principle of independence simplifies calculations and applies to many real-world scenarios. Within our problem, damaged fibers are independent events. This suggests that if one fiber is damaged, it doesn't change the likelihood of another getting damaged. Independence is important, especially when: - Using models like the Poisson distribution, where you assume events occur randomly
- Making calculations simpler, since probabilities of independent events simply multiply Recognizing independence here helps underpin the assumption needed for Poisson distribution, making it a valid tool for this problem.