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The Poisson distribution is a fundamental concept in statistics. It models the number of events occurring within a fixed interval of time or space. This distribution is applicable when these events occur independently, and the average rate of occurrence is constant. In the context of our exercise, the experimenter is observing bacteria colony counts in subsamples dealt with by two methods.

Here, both the counts for method A and method B are modeled as Poisson random variables, with their mean event rate being the same (i.e., \( \lambda_1 = \lambda_2 = 50 \)). This means, the probability distribution of the number of bacteria colonies follows the Poisson distribution.

• **Key Characteristics**: The mean \( \lambda \) also represents the variance for Poisson distribution.
• **Independence**: Counts from different groups (method A and B) are modeled to occur independently.

Understanding these characteristics helps in framing problems where random events happen over intervals, such as bacteria count in a sample.

In many scenarios, when we deal with large sample sizes, using the Poisson distribution directly could be complex. That's when the normal approximation becomes useful. For Poisson distributions with large means (like \( \lambda_1 = \lambda_2 = 50 \)), we can approximate these with a normal distribution. This simplification helps in calculating probabilities more easily.

The normal approximation takes advantage of the central limit theorem, which generally states that the distribution of sample means will be normal or nearly normal if the sample size is large enough. So, in this exercise:

• Method A: \( X \sim N(50, 50) \)
• Method B: \( Y \sim N(50, 50) \)

When dealing with the difference in counts \(X - Y\), the approximation results in a new normal distribution \(X - Y \sim N(0, 100)\). This is derived by understanding that \(X\) and \(Y\) have their variances summed (50 + 50 = 100).

This conversion into a normal distribution simplifies the probability calculation of complex Poisson problems, making it a crucial step in statistical hypothesis testing.

Once we have approximated the original Poisson problem to a normal distribution, calculating the probability becomes a more straightforward task. For our exercise, the task is to find the probability that method B is superior to A, meaning where \(X - Y > 10\).

To calculate this, we need to standardize the variable. Standardization helps shift the distribution to a standard normal curve, which has a mean of 0 and a standard deviation of 1. The formula for the standardized variable \(Z\) is given by:

• \[Z = \frac{10 - 0}{\sqrt{100}} = \frac{10}{10} = 1 \]

This transformation converts the problem into finding \(P(Z > 1)\). Standard normal distribution tables or calculators typically display the cumulative probability or the probability that \(Z\) is less than a certain value.

To find \(P(Z > 1)\), you would actually look for the tail probability in these tables, which is often approximately 0.1587. Understanding this approach ensures you can derive likeliness of occurrences, a fundamental skill in statistical analysis.