91Ó°ÊÓ

Probability is a measure of how likely an event is to occur. In the context of this problem, we're interested in understanding the likelihood that at least one grasshopper is found within a specific area. The probability here can be represented mathematically and calculated using a Poisson distribution. When we want the probability of finding at least one event (grasshopper) in a given area to be 0.99, we are essentially saying we want this event to happen almost certainly. This concept is essential in many real-world applications where decision-making under uncertainty is required. In this particular case, working with a Poisson distribution helps us express and solve the probability question quantitatively.

A circular sampling region is an area where measurements or observations are taken in a circular shape. This structure is common in various fields such as ecology, agriculture, or even quality control. In this problem involving grasshoppers, our circular region is defined by its radius, which we need to determine. Once we establish the radius, we can easily compute the area since the area of a circle is given by the formula \(A = \pi R^2\).In the context of this exercise, determining the correct sampling radius ensures that the probability of finding at least one grasshopper meets our desired threshold. Understanding regions like these can help efficiently allocate resources or plan further studies based on initial findings.

Expected value is a key concept in probability and statistics which indicates the average value one might anticipate from a random event following a specific probability distribution, in this case, the Poisson distribution. The expected value in a Poisson distribution with parameter \(\alpha\) gives the average number of events (grasshoppers) expected in a unit area, here specified as per square yard.For this problem, given that \(\alpha = 2\), the expected number of grasshoppers per square yard is 2. When the problem asks what circular region size gives a 99% chance of finding at least one grasshopper, it implies utilizing the concept of expected value to scale up the average occurrences to align with the desired probability. This calculation allows us to plan the sampling area optimally.

Natural logarithms, denoted as \(\ln\), are logarithms to the base of the mathematical constant \(e\), where \(e\) approximates to 2.71828. In the context of solving probability problems like this one, we use natural logarithms to linearize exponential equations.In the original solution, we had the equation \(e^{-2\pi R^2} = 0.01\). To unravel the exponential nature, we took the natural logarithm on both sides, simplifying it to \(-2\pi R^2 = \ln(0.01)\). By transforming the equation this way, it becomes straightforward to solve for \(R\).Natural logarithms are a valuable tool in simplifying complex calculations and are widely used in various branches of science and engineering as well as finance and data analysis.