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When dealing with a Poisson distribution, the mean (\(\lambda\)) is crucial for understanding event occurrence within a given area. In this problem, the number of saguaro cacti per square kilometer is specified with a mean of 280. However, to find the mean number of cacti in a smaller area, such as 10,000 square meters, a conversion is necessary. Convert the area first, knowing that 1 square kilometer equals 1,000,000 square meters. Hence, 10,000 square meters is a fraction of a square kilometer: \[\frac{10,000}{1,000,000} = 0.01\]Thus, multiply the original mean by this fraction to get the mean number for the smaller area:\[280 \times 0.01 = 2.8\]This calculation shows that the mean number of cacti in 10,000 square meters is 2.8. Understanding how area conversions affect the mean helps in accurately predicting occurrences within specific zones.

To calculate the probability of certain events within a Poisson distribution, it's important to apply the probability mass function:\[P(k; \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}\]Here, it defines the probability of observing \(k\) events given a mean \(\lambda\). For example, to find the probability of having no cacti (\(k = 0\)) in a 10,000 square meter area where the mean is 2.8, plug in the values:\[P(0; 2.8) = \frac{2.8^0 \cdot e^{-2.8}}{0!} = e^{-2.8} \approx 0.0608\]This result tells us there is approximately a 6.08% chance of no cacti being present in the given area. Using the Poisson probability formula for different \(k\) values can predict various outcomes, which is invaluable for statistical modeling.

Converting areas is a practical step when linking probability outcomes to physical space. For instance, how do we determine an area where the likelihood of at least two cacti is 0.9? First, calculate the probability of observing at most one cactus in this area:\[P(0) + P(1) = 1 - P(X \geq 2) = 0.1\]Utilizing the expressions for \(P(0)\) and \(P(1)\), with \(P(0) = e^{-\lambda}\) and \(P(1) = \lambda e^{-\lambda}\), find \(\lambda\) such that:\[e^{-\lambda}(1 + \lambda) = 0.1\]Solve this numerically, and once \(\lambda \approx 2.3\) is identified, convert this into an area. Since this new \(\lambda\) represents the mean for the unexplored area, use proportions to convert back:\[\text{Area} = \frac{2.3}{280} \times 1,000,000 \approx 8,214 \text{ square meters}\]This solution illustrates using mean-based transformations to map probabilities in distinct locations, highlighting the versatility of Poisson distributions in environmental studies.