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Probability calculations are essential in determining how likely certain events are to occur. In the context of the Poisson distribution, the focus is on finding the probability of a certain number of events happening within a fixed interval of time or space. Using the formula for the probability mass function (PMF),

• \[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
• "\(X\)" is the random variable representing the number of events,
• "\(k\)" is the number of events of interest,
• "\(\lambda\)" is the average number of events in the interval,

the probability of a specific number of occurrences can be calculated. For instance, to find the probability of at least 8 organisms in a cubic meter, calculate the sum of probabilities for having fewer than 8 organisms and subtract from 1. This requires calculating each probability for \(k = 0\) to \(k = 7\), summing them up, and using:

• \[P(X \geq 8) = 1 - \sum_{k=0}^{7} \frac{e^{-10} \cdot 10^k}{k!}\]

This cumulative approach helps in computing probabilities over a range of outcomes.

In a Poisson distribution, the mean and standard deviation are key parameters that describe the data. For this exercise, the mean (\(\lambda\)) is particularly vital as it represents the average rate of occurrences over a fixed interval. Given the problem's scenario:

• Mean, \(\lambda = 10\) organisms per cubic meter for 1 \(\text{m}^3\).

When dealing with a different volume, such as 1.5 \(\text{m}^3\), the mean changes to:

• \(1.5 \times 10 = 15\).

The standard deviation in a Poisson distribution is the square root of the mean, thus:

• \(\sigma = \sqrt{15}\).

Understanding that the mean gives you the expected number of events, while the standard deviation tells you how much the number of events can vary, is crucial. To see if the count exceeded its mean by one standard deviation, compute:

• \(P(X > 15 + \sqrt{15})\).

This step requires summing up probabilities for all values untill this threshold, enhancing your understanding of fluctuating data around expected values.

The Cumulative Distribution Function (CDF) is a powerful tool that gives the probability that a random variable is less than or equal to a certain value. In Poisson distribution calculations, it helps to determine probabilities for ranges of values rather than just a single outcome. For example, if you want to find the probability that there are zero organisms present, the CDF provides:

• \[P(X = 0) = e^{-10V}\]

Calculating CDF involves summing up probabilities from the starting point \(k = 0\) up to the desired value. This is used in calculating the probability of "at least" scenarios by subtracting from 1. In part (c) of the example problem, we want the probability of having at least one organism to be 0.999. The steps are:

• Ensure \(P(X = 0) = e^{-10V}\) simplifies to \[1 - e^{-10V} = 0.999\]
• Solve for \(V\) as \[V = -\frac{\ln(0.001)}{10}\]

The CDF not only simplifies calculations for these probability ranges but also provides a clearer picture of the likelihood of events occurring below or above a certain threshold. This understanding is crucial for real-world applications like determining the adequacy of ballast water discharge in avoiding ecological harm.